4 hrs a day sounds grim. No wonder you have had enough.
Racking my brains as to what public sector company is based in Canary Wharf of all places [very nosy], I assumed from your initial post you were based in Westminster. I take it that even though it is public sector, the option to work from home regularly isn't there or you would be doing it already or working from the nearest public library if space at home is an issue
So specifics. IMO £450k won't buy you a 3/4 bed house in Greenwich or Blackheath in the tiny catchment areas for an outstanding state primary school of which there are very few. Even fewer at second level. 
There are lots of small 3 bed terraced houses in the area, with teeny gardens so you could buy a doer upper quite easily and convert the loft which most of us are doing but it would mean waiting until you have the cash and living with builders for a bit which may or may not be an issue depending on how much space you are used to at present. There is lots to do in the area and lots of green open space but sadly you are unlikely to find it in your back garden. Someone will be along with a link but I've checked out what's been posted so far and it is either v close to the A2 or nowhere near a v good primary. I'm guessing that if you currently live in a nice rural village, the reality of the A2 will send you heading back there quickly.
As far as SE London goes, you could definitely get a nice house on that budget but schools will be your problem and probably should be your priority in your search. Find the school, define the catchment area and start househunting.
A new free school has opened on Shooters Hill I think as a result of poor local schools mostly which opens up Woolwich and Eltham and much bigger houses but they are rougher areas than Greenwich or Blackheath as a generalisation. It's London though - it's not uncommon for there to be serious crime in the most affluent areas. Despite the recent news from Woolwich, serious cash has been invested into the area in recent years and there will be a Cross rail station (DLR already) opening there, which can only increase the influx of people buying in the area. At present though, it is predominantly a socially deprived area.
A number of friends are moving from Greenwich to East Greater London - Epping area, Essex and Bedfordshire primarily for the schools, but with leafier feel and quicker commute into Canary Wharf than other parts of London. I can't personally comment on those areas though. Several colleagues/friends commute into the Wharf from Sevenoaks, Paddock wood and Tunbridge Wells on fast trains, changing at London Bridge. Schools are for the most part better on average out there and more bang for your buck with a house.
Chislehurst, Orpington, Bromley are the other naice 'burbs worth a look. You're still within a quick commute into London Bridge area (30 mins - probably an hour door to door all in or you could cycle), good schools and leafy feel.
Hope that helps - happy to answer any specific questions on the Greenwich area if you want to PM me.
Have you considered renting a room Mon - Thursday? Not ideal but you could discuss working flexitime with your employer. Lots of people I know are now working a 9 day fortnight - increased hours for every other Friday off. Presumably if you do a 4 hr commute you don't see your kids much Mon-Friday anyway?