UKA's latest set of proposed rule changes has the following:
"Unnumbered New Rule – Audio Equipment in Road Races: a proposal has been received from the UKA Road Running Management Group to the effect that audio equipment using earphones must not be used in road races and that any use of such equipment may result in disqualification."
I can see the elements of your dilemma. If you have a job in the City or other points East then travelling backwards and forwards to Richmond is going to consume valuable running/socialising time.
Living in Fulham (certainly the Chelsea end) you could do your mid-week runs in Hyde Park, your speed session at Battersea track, and your long runs in Richmond or Bushey Park - and not have to travel far to go out in the evening.
The suggestions elsewhere about Blackheath, Hackney and Ealing will not really match your criteria (and I'm the women's team manager in Ealing).
Competitively I would leave sorting out a club until you get here. You can't represent a club in competition until you have been resident two months anyway so there's no rush. If you're a serious competitive runner the first priority would be to find the right training group, a lot of the best ones have members from several clubs.
Richmond would be a better all round choice for living and running than the Docklands. The Docklands is too "inner city" to have many running options on your doorstep though safety wouldn't be a big issue.
If running territory is your main concern then the best choices are based around the big parks:
Richmond: great running and a lively place - also close to Wimbledon Common another great running park but best to run in company as it can get quite isolated there
Teddington: across the river from Richmond, real hard core running based around Bushey Park. This is where the Kenyans base themselves - not lively.
Hampstead Heath: another big running area but hillier. Lively and close to town.
Hyde Park: the central location for running. Live in Kensington or Notting Hill to get close (expensive).
Epping Forest: out in the North East, miles of running but quiet and isolated.
If you're going to be working in the City or Docklands then the commuting times might favour the more central locations - but if you want to train in a strong women's running group then Richmond and Hampstead would be the choices.
A great resource for working out how long it takes to get from A to B in London is http://www.tfl.gov.uk/tfl/. Just put in the location of where you will end up working and where you live and it will give the commute time by your chosen method of transport.