Ceramics.np.04

Good to know they are all the same besides the hairpin length. It can be tricky to get an exact read on these things.
 
Having them all at 2.2 ohms means the kiln is drawing just over 10 amps on 240v supply  so I am not surprised it wont reach top temperature/get there slowly. I would maybe confirm it with whichever company but the basics should be voltage/amps = resistance so 240/15 = 16 ohms for 10 elements in series and 1.6 ohms for each element.

@High Bridge Pottery I've measured the resistance and they're all the same (sides and bottom), around the 2/2.5 mark... the number kept fluctuating quite a bit but 2.2 seems to be a thing(!?) 

Suggest: Just count the number of complete coils in the entire length, measure the length. This can be done on several just to confirm they are the same or not.  Euclid can then distribute evenly once a wire size and resistance value is picked.

The bare copper jumpers are an odd way to do it, and not necessary. Instead, just leave the element pigtails long, bend them to meet up and overlap about an inch, and put a connector on them.

They look about the same to me even though some are not very even. Lets hope they are all the same resistance and the floor just have a bigger hairpin. Do you have a multimeter to test the resistance of each element? You would need to turn the dial to select a low resistance setting. Depending on the type it could be 200 Ω or lower like 20 Ω but the lower the better as they will only be measuring around 1-2 ohms.
I have drawn a basic picture to give you an idea  You will have a black and red probe and put one on each leg/tail of the elemet to get the value. Make sure you turn off the kiln isolator switch first.

 
Still not sure what that black box is doing, I thought maybe it is the relay but then the controller has the relay inside so I guess it is another kind of safety switch? I don't think the bare copper is a problem but as Neil says you can just connect the element tails together with a connector if they are long enough.
 
Edit: maybe an actual picture could help 

@neilestrick @High Bridge Pottery At the risk of being the cause of a few exasperated sighs and head shakes: most of this is going completely over my head. I am completely out of my depth...but also stubborn and ready to learn.
I was worried you were going to say that. Yes, it is just bare copper (which even I thought was a bit odd) and, once I'd taken the cage off the back and cleaned it all up a bit, everything about it looks pretty old and tatty. 
I've tried to take some photos in sequence so the wiring might make a bit more sense:
        
 
and then some close ups of the two...er...connect-y thingies:
    

This is brilliant. THANK YOU! 
Could cracks in the fire bricks cause the firing to slow down significantly? They don't seem to be going all the way through but there is a pretty big one running across the top and I've just found one down the back when I took the cover off to take photos 
Photos of the back:        
 
...and cracks:       
 
I have some high temp cement to fix these. Would it be worth trying before getting new elements or will it not make that much difference? 

You could ask Euclid's if they would wind them for you. Send them all the info off the faceplate, kiln measurements, photos and any other info you have and see what they have to say.
https://euclids.com/pages/kiln-elements

I am surprised pottery crafts said they wouldn't do it, probably just being lazy as they don't have the specification on file.
The few places I know who would probably wind them for you are
https://www.kilncare.co.uk/elements
https://www.kilns.co.uk/kilns/kiln-elements
https://www.cromartiehobbycraft.co.uk/Catalogue/Ceramic-Kilns-Electric-Kilns-Pottery-Kilns/Kiln-Spares-Kiln-Elements/Kiln-Elements-Element-Wire-Connectors

@High Bridge Pottery Thank you so much for this. So helpful...and makes it all slightly less overwhelming.
My kiln does have four elements up each side and two at the bottom, with the bottom two having a bigger hairpin (...is that right?!) than the side ones. 
I did get in touch with pottery crafts and they told me I would have to find an element winder to make them for me specially. Am I right in thinking I can send the coils to someone for them to make something similar? Would I just need to know the correct voltage, if so? Thanks again!
 

1300*C is roughly cone 12, if we’re going by the slower rate of 60C/hour in the last hour. Cone 10 is 1285* C, and and and cone 6 is 1201*C.
In North America, if a kiln is rated to cone 9 or 10, it means it will hit that top temperature for a handful of firings and struggle to do it after that. A cone 10 kiln in all practicality here is a cone 6 kiln. However UK electrical standards are very different, and I believe the standard thermocouples used in the 2 regions are also not the same. But I’d venture that if your kiln is rated for cone 12, with proper replacement parts you ought to be able to reach cone 9 or 10 regularly. Ask some questions of the person you buy your elements from. 

Hi Ceramics.np.04,
Above I'm disagreeing with PeterH, looks like your kiln is rated Cone 10 (or more), not Cone 6.
I should have been more specific.
A Cone 10 kiln element set should last a good long while when used to fire to cone 5/6.
I'm just saying it looks like a Cone 10 (or more) rated kiln.

I would say in the UK 1300c and cone 10 and are pretty much interchangeable even if most kilns wont manage 150c/h for the last 40 min of the firing up to 1300c.
 
 
I would test what voltage you have in the studio over a few days and get the elements made for that voltage. The UK is supposedly 230v but I get 245V at my house so any elements designed on 230v will draw more power on 245v. Would be worse the other way around running 240v elements on 230v as then you will be down on power.
Just realised it is a front loader which tend to be more complicated than top loaders. Just a guess at the elements inside being 4 in the side walls and two in the floor. The side wall elements may all be the same spec but the floor will possibly be different.
Need to know the length of the tails (probably double what they are now so you have extra to play with/cut off, maybe 9-12") what length each bit of element is in the wall and how long each hairpin is. Also need to know what gauge the wire is, the internal diameter of the coil (maybe 1/2 or 3/4") and how they elements are wired up. Once you know how they are wired up we could possibly figure out what resistance they should be.
A bad drawing to maybe help.

This makes a lot of sense. I hadn't heard of this but, judging by the age of my kiln (and it looked pretty well used when I got it), it must have done a fair few firings. 
I knew this day was coming...I was just hoping I'd be able to get christmas out of the way first 
I'm going to give it a go, monitor the temperature and the timings and hopefully end up with something giftable... 
Thanks again for all the help!

Even if you just get the little glass plate that goes in the welding visor, that’s better than going without. 
If you’ve got to have the pieces for Christmas,  it is possible to do it, but it’ll be under less than ideal conditions for your kiln. The good news is, once you replace your elements, and likely the relays too, you’ll have essentially a brand new kiln. Insulating bricks don’t expire. 
 

You may find this of interest, especially the sections What is Heatwork?  and  Put Witness Cones in Every Firing.
Temperature vs Heatwork – Why We Use Witness Cones
https://suemcleodceramics.com/do-you-put-witness-cones-in-every-firing/

Hey just a late add here, you will be looking to protect your eyes from infrared light, not necessarily ultra violet. The lowest shading coefficient I have seen is 3 with a range of 3 to 6. If you buy reputable welding goggles or even buy from a reputable kiln supply house then this should not be an issue. I have on occasion met folks who bought UV glasses, more sunglasses however so worth mentioning that you take care and purchase proper lenses. For kilns you will be looking into an extreme infrared environment. 

Thinks for posting the kiln plate. I hope that the experts will chime in, but I suspect that the max temperature of 1300C means that the element life - when firing to your desired temperature -- is likely to be quite short.
For example see the first answer in
PS I'm not certain if the figure of 50 firings includes the lower-temperature bisque firings.

Cones are easy,
If you want to follow the cone chart then for your last segment go 60c/hr in the last 100c of the firing to 1222c. So
pick a cone, let’s say 6. find the temperature in the 60c per hour column = 1222 subtract 100 from it, 1222-100 =1122 program your final segment to start at 1122, go 60c per hour and end at 1222. the last 100c is super important to the heatwork done. From the times you state though it does not appear your kiln is keeping up with the program and therefore likely worn elements.
For a worn kiln you may be able to use the 15c column which means for cone 6,  your ending temperature is 1185c, final segment is 1085c => 1185c and final segment firing speed is 15c per hour.
There is a limit to how slow you can go and how glazes perform.  
As far as cones and many other things, In the final segment It’s rarely only about top temperature. As an example, just like baking something, it’s not simply 170c, it’s 170c for 30 minutes ………. For kilns we depend on rate per hour though.
If you cannot find elements Euclids.com can likely wind new and they ship worldwide.

As someone who’s been firing an analog kiln outdoors in the Canadian winter for the last few years, I can confirm that -9 is not enough to make a difference in your firing speed. I don’t start getting much slow down until about -25 or so, and even then, it’s only about a 15 minute difference.
Since your elements are likely worn, you can still continue, but definitely include the cone packs this time, and make sure you’re checking the firing every 15 minutes for the last 2 hours or so. You want to be able to turn the kiln off yourself once the cones fall.  If you fire overnight to avoid peak electricity rates, maybe start it about 2-3 hours later than you would normally, so you’re awake for the end. 

Yes definitely possible. So silica and alumina do not melt basically on their own below about 1600c/3000f. So they need flux to help melt at a lower temperature The fluxed reaction pretty much takes place in the last 100c of the firing, prior there is just not enough energy to get things going. So temperature and firing speed in the last 100c is super critical, to which PeterH has posted the Orton cone chart above.
As kiln elements age their power decreases and eventually they can’t make a reasonable final firing speed and things begin to over fire. The same goes for anything that slows down the firing such as worn elements AND a cold external firing temperature. In the end though -9c is cold but not that cold so likely the elements are fairly worn. How many firings are on them and to what cone do you routinely fire?
 
 
 
 

A cone temperature chart confirms that this is a real effect. A cone number relates to an amount of heat-work and the maximum temperature needed to achieve this depends on how fast the kiln is heating towards the end of the firing.
https://www.overglazes.com/PDF/Orton-Cone-Chart-C.pdf


Did you previously fire by observing the cone drop or just use it as record of the firing?

Stumbled across this while looking for something else, don't know if it is of relevance .... at least it's UK.

Kiln Elements, Repair, Servicing and Support https://www.potterycrafts.co.uk/Products/kiln-elements-servicing
To get a quote on your elements please contact us with full kiln details as specified above. We are also able to match elements from other kilns if the old elements are returned so that we can measure the wire, coils and length.