First of all I wanted to say how sorry I am to hear about your wife. You have a lot to bear, especially with young children. If you live in Herne Hill or Dulwich you would triple your access to train lines as you would have London Bridge from North Dulwich station, and Victoria from Herne Hill as well and Farringdon, City Thameslink and King’s Cross from Herne Hill. The overground at Denmark Hill is also a short bus ride away. That would get you to Canary Wharf. I think with your budget you would easily find somewhere to rent and then you can reassess after a year. I looked on Rightmove and there is a cottage on Dekker Rd which is Dulwich/East Dulwich for well under your current rent. Good luck!