What is your £650-700k budget based on if you don't mind me asking? How much are you expecting to be able to borrow from banks - do you have a mortgage in principle? Be mindful that lenders are now beginning to impose restrictions.
Seems to me that you are happy renting in central London (Though £2k is obviously a high rent and you could pay a lot less with a mortgage), guessing you are renting quite close to your place of work, and am I wrong in believing your place of work may be near Gower St
? I only ask as I used to live quite close to Warren Street station and I absolutely loved it there. I also enjoyed exploring the local areas (Fitzrovia, Camden, Chalk Farm, Hampstead etc) and knew I'd rue the day I had to leave. Since London is new to you, perhaps you ought to aim to stay quite local - you seem as though you don't want a huge upheaval for your young son and that's understandable. Camden, Chalk Farm, Belsize Park, Tufnell Park, Hampstead are all good options. There are lots of new build flats in these areas too, flats that were massively overpriced which will likely come down in price soon, so you may be able to bag yourself a good deal. There was a thread on here recently on new build properties, and a lot of the posters were claiming how happy they were living in one, as energy bills are much lower, and there are fewer issues to deal - though do be mindful about service charges.
Islington/Cannonbury are also good areas to consider. Very liberal, nice high-streets, good community spirit and a very easy commute to Euston - you could even walk! I'm not as well versed on Finchley and surrounding areas as @Desiringonlychild but I've heard good things, and being on the northern line would mean an easy commute.
If it's a 2 bed flat you're after, and you want to emulate your living situation when you were in Europe, my suggestion would be to forego the house, and opt for the flat. You'd deffo get more for your money in Walthamstow though (North London is expensive for what you get compared to south imo)- the Victoria line starts in Walthamstow so your commute to Euston would be easy peasy. It's very liberal, lovely village and I know people with families who live there and are really happy.