BA have been operating LHR-LCA too using a 787 so there's obviously some demand. I haven't seen A3 operating every day but just Friday, Saturday and Sunday.

The British military presence in Cyprus is generally taken care of through Air Tanker A330s which operates Brize Norton to Akrotiri and that still seems to be operating as normal.

I guess there are plenty of Brits living in Cyprus on both sides of the border and at the moment the Turkish side is totally locked down air traffic wise.