What keeps BA (and VS) at LGW is almost certainly directly linked to capacity constraints at LHR.
Had LHR been adequately expanded when neccessary (back in the 1970s/1980s) and BA (and VS) had the required slots and terminal capacity that they need, it is possible that they would not be at LGW at all. Under such (fantasy) circumstances it would make no sense to have a base at LGW with all the extra expense involved.
LGW brings no extra benefits that cannot be had at LHR, except some slots and some terminal capacity. LGW is an "overflow" for mainly "point-to-point leisure routes.
LCY on the other hand is quite different: it is a mainly business "niche" operation close to London's two financial centres, and has a unique selling point of speed and convenience for London businesses (e.g. short check-in times, no long trek out to LHR, etc.).
The situation with BD is unrelated. That appears to be a legacy of rubbish management over several years, and the demise of that carrier is tragic.
Some of the acquired BD slots will be used for new much-needed longhaul routes at LHR, not bringing parts of the LGW operation accross. If there has to be a LGW operation, it needs to be of a reasonable size, (not least to benefit from economies of scale). There is no point dragging part of it to LHR just leaving a rump at LGW, especially if they are to give U2 a "run for their money" on shorthaul.
Suspect that BA (and VS) will remain at LGW for the forseeable.