75Bedford - Before you go spending money on private tests, I would suggest you have to be clear what your expectations are when you have the results.
Both my partner and myself have had bloods done overseas, and whilst mine were interim diabetic panels (when overseas for a very extended period), MrB's were in order to fast-track something when he returned to the UK.
When MrB presented the results to the Doc, she wanted to run bloods of her own, rather than rely on a lab she was unfamiliar with. Of course, the additional factor there was the anaylising lab was overseas, but I'm guessing even if your Doc is happy to look at them, he/she could be a bit put out that you'd doubted his/her tests/treatment.
I'm not trying to dissuade you from having the tests done, just would hate you to spend your cash, only to find the Doc to be dismissive.