The problem here is NOT the fact its private or the type of birth she wants.
The problem is she sounds like she is high risk (I think being 17 in itself increases some risks so its not just her BMI).
Who will be giving her antenatal care? Will she do this locally? Or will it be combined between London and Gloucestershire? If its a mix it increases the risk of something being missed or crucial information being passed on. If there is something not quite right what would she do? Would she wait and see because she didn't want to go locally, go to London to get checked or get checked locally? This is a very important question.
I did consider going private in London for various reasons despite being 180miles away. I didn't.
The reason was it was irresponsible and to do so would have effected my decisions had there been complications during the pregnancy which would not have been in the best interests of the baby.
You could rent somewhere in London for a short while, but realistically you'd only do it for a short period and certainly not from mid way through her pregnancy. It might not be necessary but you just don't know that.
And in a premature emergency situation, being forced to go locally when she is unprepared and unfamiliar and doesn't want to go there, could make a traumatic event even more so, which is not good for either your daughter or grandchild.
I honestly think the best course of action is to flag concerns she has with midwives locally and to get them to build a relationship with her. As she is 17 they probably will be willing to spend more time with her. If she makes a point of saying that she has been treated badly previously on the basis of her age and weight, and she isn't prepared to tolerate it then the way she is treated may differ as a result (sad but unfortunately true). Being assertive and having an advocate makes a difference. It doesn't mean that it will eliminate being treated badly or having a bad experience, but it can help especially if you highlight the fact it is causing a significant amount of anxiety.