I've searched through the Kennel Club website and can't find any answers so...

How do you go about getting permission to show following an operation? Is there a standard form to fill in like there is with neutering?
How long does it take for permission to be given?
Do you have to give the letter to the judge every time you show, or only if they ask?

It's for a tooth removal btw.

As far as I know you just send a letter to the KC explaining what the operation was and a letter from the vet explaining the reason for the operation and the fact that it was required because of an accident if indeed it was.

If the site of the op is obvious I would carry a letter and tell the steward you have permission to show.

I did this a few months back, just sent a letter in to the KC with a letter from the vet saying what had been done. Mine was the same as yours tooth removal. I had a letter back within two weeks.
As  far as I am aware you do not have to show it to anyone, unless they ask. So I would just stash it in your show bag in case it is required.
My dog had to have a back tooth removed due to trauma, but I doubt very much if a judge would notice, I have being showing for quite a number of years now and I have never know a judge look at the back teeth they have just checked the dogs bite.
If it is a front tooth that has been removed, then that would be a lot more noticeable and so you would probably more likey to be asked about it in which case you could say you have permission to show, and can produce the letter.

It hasn't been removed yet but she's booked in, it's an upper carnassial so doubt a judge would even notice.

Which breed? If your breed is not one where full dentition is normally checked, the judges probably won't notice. Some breeds even allow teeth to be missing due to working injuries, so always a good idea to check your breed standard and see exactly what they say about teeth. Knowing which judges are "tooth fairies" is always a good idea as well, so you know not to bother showing to them.

Doesn't matter what the standard says about teeth or whether judges look or not, any operation that alters the confirmation of a dog (which removing a tooth does) must be reported to the KC and permission asked for.

My oldie lost one of her big canines and not ONE judge ever noticed in the past couple of eyars she was shown, not even the continental judges that check so carefully LOL.

Oh I wasn't saying that permission shouldn't be requested, was just commenting on the fact that the missing tooth probably wouldn't be noticed by most judges if the breed doesn't require a full dentition check. And the reason I said to stay away from the tooth fairy judges is that some judges will penalize dogs with missing teeth even if the dog has permission to show.

I have written a letter myself and had the vet write what was done.

I have three letters in my show bum bag (in fact one of the dogs is deceased).  I have yet to be asked for them. 

I have put them in a polypoclet and then sealed the top and folded up, so it stays intact.

The two still relevant are both veterans and both for teeth (one upper carnasial removed due to abscess, and the other a front one broken by an epulis), both have won CC's and RCC's since.