I'm not entirely sure that I've grasped the whole of your question properly.
However, as I read it, you want to start from the London area, travelling to a destination in the UK or Ireland and returning in little more than the time that it takes to get through the arrivals part of your destination airport and then back through check-in.
If so, an obvious choice of destination (simply because there are so many flights going there from the UK) is Dublin. For example (using a sample date of 14 November( you could fly out from London City with BA at 0655, arriving in Dublin at 0820. You could then return at 1055, arriving back at London City at 1225. There are other flight combinations which you could use as well. Further, you don't have to fly out and back with the same airline. CityJet also serve that route, giving you additional options.
Flights with BA and CityJet could also be combined to travel between London City and Edinburgh.