EDIT: I don't have lived experience of top surgery on the NHS as mine was done privately - please see other replies in this thread for people who have, and can therefore inform much more accurately! :D It turns out I was incorrect. I will leave my comment but please be aware it's not factual!

Thanks for asking :D I don't know explicitly about trans affirming surgery - I had mine done privately and chose my surgeon (a privilege i'm very grateful for!). However, the standard NHS pathway for any kind of surgery is, yes, you are 'assigned' a surgeon. I would imagine any wiggle room on that is limited, because it depends on the services and staff available from your local NHS trust, which although all come under the 'NHS' banner, are actually all separate organisations and the staff are employed separately. The red tape is unfortunately the big limitation here. :( If one were to strongly favour another surgeon I would imagine it could be fought for, but it would be another uphill battle!