Hi OP, sorry you haven't had much in the way of a response yet. I don't know Wren Finchley personally, but I do know a couple of members of staff there, and I'm familiar with its sister school Wren Enfield, which is run along pretty much identical lines.
I think both schools will always be a bit 'marmite' and you definitely need to be signed up to the ethos. They are undoubtedly quite strict/rigid (what I'd describe as 'high control') and have a fairly narrow curriculum.
It's not for us, and we're actually moving area partly so that DD doesn't have to go there (her current primary is a feeder for Wren Enfield) - we want something academic but a bit more holistic, with more of a priority given to creativity, critical thinking and the Arts. We've done a tour and found it all a bit joyless, the staff overbearing and the buildings/site cramped and a bit depressing (but again that's Enfield not Finchley).
On the flip side, Wren Finchley's results are undoubtedly excellent for a full comp, the families who buy into it seem to love it and those I know who teach there say that lessons are calm and orderly due to the strict discipline. Where I am there are far fewer behaviour issues with Wren pupils than most of the other local schools after school (anti-social behaviour is an issue with many of the other local comps).
Sorry it's not a deeper insight but I hope it's vaguely helpful and you get some more first-hand responses soon!