Yes I think you are going to struggle to be 30 mins commute for you both within even £500k budget in a nice area, it will definitely be a flat (poss with garden) rather than a house/maisonette at that price - nothing wrong with a flat of course!
Out of interest why the prohibition on East London as that would be the most convenient direction for your work - most of Hackney for instance is very trendy and middle class now and would be a super easy commute to Liverpool St, slightly less so to Kennington but you could do train to Liv.St, either walk/tube to Moorgate and northern line from there?
You could also consider going a little further out northwards from where you are now, Manor House/Finsbury Park is an area that's been gentrified a lot in the last few years (the areas around the stations still not great but once you are 5-10 min walk away lots of nice quiet streets of victorian terraces), great transport links with the choice of piccadilly or victoria lines plus some overground stations as well. Budget will be a stretch but you should be able to get a small 2 bed period flat (probably no garden) within a mile of the station...
If you can stretch the commute to more like 40-45 mins, and can get over your tubes vs trains hang up, you will get a lot more for your money if you venture south -east wards, areas like Forest Hill, Hither Green, Lewisham, Blackheath are very nice and very popular with families. I understand the train fear but this close in to the city the trains will be nearly as frequent as the tube (maybe not every 2 mins but likely 4 or 5 an hour) and delays are not really a big issue - I am betting your colleagues who moan come in from a lot further away. You should be able to get a nice 2 bed garden flat, commute would be OK for both going into London Bridge where you can pick up northern line southwards for Kennington and northwards for Moorgate which is 5 mins from Liv. Street.
If you really want a 3 bed place then I think you'll need to go more to zone 4-5 which probably means closer to an hour's commute door to door for one of you at least, is that an option?
As to when to move, ideally I think move before baby, particularly if you want to do lots of work to the place. But then again the purchase process can take a long time in the UK, expect 3 months minimum and 6 months is not unheard of. Lots of London people I know have babies in 1 bed flats so it wouldn't be a disaster to stay where you are now (baby sleeps with them for first 6 months at least so lack of 2nd bedroom not an issue there, storage can be a problem though)...