NHS - completely free, but you can pay for an "amenity room" (i.e. your own room on an NHS ward) . These cannot be pre-booked as a rule so it is luck of the draw whether you get one. Costs in London hospitals range from the piddly (£25 pn I believe at St Georges) to expensive-ish (£150pn 3 years ago at Queen Charlottes). Insurance will not cover this cost. Depending on the hospital you go to you might be able to get a private room on a private ward after an NHS birth. This would cost you £400-£600pn (I think - it's 3 years since I gave birth) and again insurance won't cover. I don't know if you can pre-book. Basically these are just ways of getting your own room.
Private hospitals or private maternity wards in NHS hospitals (Portland, Lindo, John & Lizzies, St Thomas', Queen Charlottes, C&W) offer various services with different costs. A midwife-led delivery at the Portland or J&L seems to cost something like £4k, but if you wanted extra nights in the hospital that could be pricey (£750 pn at the Portland the last time I looked). If you want or need the services of a consultant you can add a few grand on top. I suspect that with a consultant-led birth at one of these you would be lucky to get change from £10,000.
You don't say what sort of insurance you have but if it is a standard UK policy it will only cover things related to childbirth which are medically necessary. So you will only be covered if, for example, you have a planned CS as a result of, for example, placenta praevia or an emcs. Even then, the insurers often don't cover absolutely everything so you end up paying something too. There is one provider that I know of which covers a normal birth but my friend told me it cost over £1000 per month!!
You may well be too late to book at the Lindo, and possibly J&L, but the Portland might have space if you feel strongly you want to go private. They seem to be able to book people much later on (judging from my obsessive viewing of Portland Babies on the TV!).
My own view is that the NHS provides a good service. There are not many frills, but care is there for you as and when you need it - you aren't for example, encouraged to be induced on a day your consultant happens to be doing private work, as happened to a friend of mine who went private. And if you have a straightforward delivery you would be discharged very fast anyway (to the care of community midwives - you are not sent home with no further contact).
I also rate the NHS because I would hate to be ill in the hospital and worrying about costs racking up. Or to have an ill baby in the NICU and to be fretting about the expense. If I had a sick baby I would want to be on the NHS for sure.
A halfway house is to use an independent midwife to provide continuity of care and to act as doula at the NHS hospital. I'm doing that this time around and it is working well. And when I am discharged someone I know and trust will provide post-natal care for 6 weeks which is just great.
HTH.