I would advise you to think long and hard about doing this in Switzerland, because it will be a bloody nightmare to have to deal with going forward....
I knew two people who did this - an Irishman and a French man. Since the concept is unknown here, every little thing to do with administration became an issue - why does this document not match up with the other, you must have stolen the documents, you are an imposter.... in the case of French man, he ended up getting a visit from the police because the credit card company decide he was attempting to commit fraud by applying for a card in someone else's name.
Once you have the deed, it is very important to have the document apostilled. This confirms the signatures being applied to the document are authorised and valid, and Switzerland is a part of the Hague Apostille Convention so the document is valid in Switzerland as well. I assume also you will have a new passport and driving license in your new name? Your old passport will then be cancelled and no longer valid.
The local commune has to accept this, as it is now your legal name, and you are a UK citizen. If speaking to credit card companies or banks you may need to show the document and explain what you have done.
Once you receive the original apostilled document it's a good idea to make several copies and have them certified as a true copy of the original by a Swiss notary. In this way if anyone asks you can send them a certified copy. However your commune will likely ask for and may keep the original, but they may also be happy with the certified copy and sight of the original. That will vary from one commune to the other. You could also ask for two originals from the deed poll office or the court in the UK.
Edit.
He almost would not have been a Swiss citizen at the time of that change. That could make it easier. The name on a UK birth certificate is unalterable so does not respect later changes.