I've had an independent midwife both this pregnancy and last. Last time I booked her at 19 weeks because I was refused an NHS homebirth and was coming to realise that I needed more support than the NHS could provide... this time I booked with one right from the start. So there's no need for any NHS involvement at all... however, women booked with independent midwives retain the same entitlement to full NHS services (community midwife/consultant care, scans, any medical investigations, medication, hospital delivery if chosen/needed etc), within the normal constraints of what is available for any given postcode.
Each time it has cost around £3,000 (but prices in my area vary from around £2500 to £4000, so it's worth asking around if your budget is tight). Each I've paid privately for scans but stayed with the NHS for medical care and investigations from non-obstetric consultants. Most of the way through I've found NHS staff very willing to work alongside independent midwives, and indeed appreciated the way in which independent midwives had more time to give us than them. (The only exception was one health visitor afterwards who somehow decided we might be using non NHS services as a way to cover up evil doings... but she was very much an exception, and one which coincided with the unfortunate sagas of Baby P.) So whereas the systems have coordinated well for me, I've sometimes needed to have been quite proactive and confident in getting the different people involved to communicate.
So if you still wanted to book with the hospital, you could choose either to take an independent midwife with you to consultant appointments, or ask her to try and liaise on your behalf. Whether the latter were possible would still depend on whether the consultant would allow it, but you lose nothing by asking.
But really, whatever you decide about your birth and however your pregnancy progress, I'd say having an independent midwife is definately and comletely worth it.