The other way around: you want to be on a through ticket. Problem is: fares between different airlines are not always combinable and cannot always be issued on a single ticket, hence why it is often more expensive (you have to have fares which are combinable for them to be on same ticket, and these normally are more expensive than the cheapest fares). There also are issues about issuing multi-airline e-tickets, when there is no e-ticketing agreements between the airlines.

Frankly, if you are on different airlines, find out what the rules are before confirming the reservation or, better, ask your travel agent to find out for you and then you can decide whether you want to run the risk or not (add padding in your schedule if need be). It looks to me like your travel agent was at fault in selling you an itinerary separate tickets without warning you of the implications of this.

It is not impossible for the information to appear on the screen. I think, although I am not sure, that it is possible to have two different tickets on a single passenger record (which is what the airline sees on its computer). In any event, it is always possible to annotate a passenger record so that the airline knows of connecting flights.

One last thought: does it say on the BA e-ticket receipt that it is 'in conjunction with' followed by the VS ticket number and vice-versa? (conjunction tickets are treated the same as if they were a single ticket for most purposes as far as the passenger is concerned). If not, these most probably are separate tickets if they have wholly different numbers.