Hi James, welcome to the forum
The marching around Bangkok (mine was a very sedate march, more of a stroll really!) would be for people who need to supply the Thai government with the documents they require in order to marry in Thailand.
You don't need those.
As for registering the marriage in Thailand, I don't think you can. That would be the same as registering a Thai marriage in the UK. Married is married, you can only do it once, in one place.
What I suggest you do is to obtain a Thai translation of your UK marriage certificate and take that to the Amphur where your wife is registered and have her surname officially changed. Part of that procedure notes the fact that the name has been changed due to the person adopting their spouse's surname.
That will result in her having a new Thai ID card issued, a change in the Tabien Ban, and the possibility of her having a new Thai passport in her married name.
This is the part that can, potentially, cause some difficulty on the return leg of the trip as her ticket will have been issued in the name on her current passport.
In order to avoid those issues I would advise that, on this occasion, she change the ID card but not the passport.
Travel back to the UK and at a later date, contact the Thai embassy to have a new passport issued with her new surname. Again, that presents some potential difficulty with her visa being in the old passport, and this has been discussed at length here before.
I seem to remember that she can keep the old passport with her visa in it in case she needs to show it to anyone.
I may be wrong about that part
The other, more expensive, option would be for her to travel to Thailand on a one way ticket, change everything (passport included) and then book a one way flight back to the UK in her new name.
As for the translation, you can get that done here. I believe the Thai consulate in Hull offer that service. Make sure that your surname is translated to your satisfaction, it might be an idea for your wife to decide exactly how it is to be written in the Thai script. She should make a note of it and make sure that it's the same in all her Thai documents.
Good luck with it all.