If you move house, you will worry about Sod's Law - would it not be just your luck when they hire a new, wrong Head or Ofsted rates it Requires Improvement out of the blue and turmoil rips it apart with staff leaving in droves etc... Or the catchment shrinks that year and you had not moved quite near enough. Something bad like that could happen - even a 10% chance is scary.
My guess is that for at least one of your kids you would succeed in either getting a place at a great state school (say partially selective or perhaps even have a go at QE in Barnet?) or a great independent. For the other one NBH may also work out OK. (Moving for 6th form could be the plan in that case).
There is a lot of "option value" in not moving and instead waiting until you see all application outcomes before you write a big cheque. During that process, you will also learn a lot about schools and how well they suit (or not) your kids. What you thought or preferred when he was 8 or 9 may not be the same at 11. Choice has value.
Otherwise, although NBH (Hampstead/Rosslyn Hill) has its issues (especially turnover), we will definitely apply there for DD2 who is 9 - but hoping she will also get an offer somewhere better, like DD1 did.
We (also) really don't want to move, and are not happy with local state secondaries - but will watch them carefully and may change our mind. We also went for state primary and love the multicultural and social mix. But DD1 (who is at a good but not superselective independent secondary) now has great, positive, hardworking friends/peers; so I would not worry much about ethos. And as we hoped, their attitude is rubbing off on her. That culture is very different from what we perceive and hear about at our poor-to-mediocre local state secondaries)
Stamp duty would feel like a huge waste, it really is money down the drain which you don't get back. BTW, I highly recommend a large offset mortgage (with lots of "slack") to help sleep well at night if paying fees.
Final point: With bright kids, you should look primarily at % A+ at GCSE rather than % A-C. (A-C becomes meaningless for all good private schools - with all at 100 or 99 it tells you nothing, and some state schools focus a bit too much on getting Cs). And when comparing state school results, look at the stats for high and mid achievers, not just the headline figure.
Good luck!