A couple of things to consider (not had BC but mum aunt and gran have and I have worked in both NHS and private hospitals).
A private hospital will be more comfortable, better food and private room.
But, the NHS hospital you will probably be on a ward in a bay with other women who have had the same treatment/op.
In a private hospital you will have nurses who are general surgical nurses, the NHS may (not always) have specialist BC nurses. When my mum had her first op one of the nurses was quite happy to show the results of her own mastectomy and reconstruction, she had decided to specialise after her own treatment.
My mum made friends in the NHS hospital who she met up with again in her support group and hospice.
Personally I would prefer to go private, I'm not the type of person who wants/needs other people to compare notes with, my mm is.
BUT before you go in to your local private hospital check who they employ 24 hours a day. Legally they do not have to employ anyone with medical qualifications, generally they do, but over night they usually don't have the staff you would in the NHS.
So although at your out patient appointments you will always see your consultant not 'one of the team' if there is an emergency or you need to see a Dr in the middle of the night or just when your own consultant is not about then you will not have access to either your consultant or one of 'their team'. At one prestigious private hospital the Dr who is there 24 hours a day (yes singular, just the one) was an ob/gynae Dr because the hospital had a maternity unit.
My preference would be to go private on a private ward in an NHS hospital or a private hospital that undertakes work for the NHS. This way you get all the nice things about the private treatment but if you need to see someone at 3am then there is someone a short walk away, not at home in bed needing to be paged and to drive in.