Ordinarily, BA flight numbers give some clues as to the route being flown; for instance, I think all flights to the US have an odd flight number and those headed back to the UK an adjacent even one. In particular, there are various assigned flight number ranges to distinguish things like codeshares, charters and the various London hubs. In the context of this question, flight numbers BA2530 to BA2899 normally denote LGW short haul operations.
However, during the pandemic BA stopped all such operations at Gatwick apart from some domestic feeders. Some flights were cancelled outright; others were moved to LHR. It was convenient at the time to keep the flight numbers the same, for a variety of administrative reasons (it was still a lot of work, some of which I was involved in - nothing is simple in aviation IT). Subsequently, some - but not all - were renumbered into the LHR short haul ranges (although at this point you start to get into Ship of Theseus territory when wondering if this is 'the same flight').
BA shorthaul operations at Gatwick will not resume until the summer season, via a new subsidiary called BA Euroflyer (which I would expect to use the LGW ranges). But for now you can be confident that your Geneva flight is indeed operating from Heathrow, despite historical use of the flight number from elsewhere.
...except as per the update, this particular flight is now cancelled. A few commenters queried the desirability of so many flights on the same segment so close together - clearly, in this case it proved not to add enough value to the schedule to justify the use of an aircraft. It just took time for this particular route to have been looked at in detail - all the LGW flights would have been moved in bulk months ago, and in these turbulent times it's best to wait until relatively close to departure before making the call on the viability of a particular flight op. In ordinary times, the 'use it or lose it' slot rules at congested airports like Heathrow prevent too many such tactical cancellations - but these are not ordinary times.
Nonetheless, such arrangements that may seem undesirable in isolation should be understood in context - to move one of the flights later from Heathrow would alter its arrival time in Geneva, and perhaps there isn't a suitable landing slot there to support such a change. Or despite arriving at the same time, the aircraft may leave Geneva at different times, with enough distinct demand for those flights to warrant sending two planes from London rather than consolidating to a single (possibly larger) aircraft which could then serve only one return time (disrupting passengers on the other). This extends to each aircraft's schedule for the whole day, as well as the movement of flight and cabin crew, who need to be in certain places at certain times. All of this is a network optimisation, which can result in individual pieces that look strange but ultimately contribute more to the whole operation than some other schedule configuration.
As for the curiosity of two flights on the same segment having exactly the same departure time? Well, carefully choreographed special events excepted, it's unlikely that two such flights will depart at precisely the same moment even with multiple runways available, but they can still be scheduled in this way. Indeed, two competitors may well settle on exactly the same flight time between two airports as a temporal example of Hotelling's law.