It occurs to me that it will be worth asking the travel agent before you make the initial booking whether they will be happy to do this post-purchase upgrade for you. While it's easy for us to predict how a paid upgrade will work if you buy a BA published fare, once a travel agent is involved it becomes possible that what you're buying is not a published fare but some other fare that comes with different rules. In theory, those could include a rule that prohibits paid upgrades. You don't want to have any unpleasant surprises after you've booked the economy ticket, and so I think it might be worth having this conversation with the travel agent in advance.

If you are doing this through a travel agent, there may be no magic in doing this within 24 hours of initial booking. If you book directly with BA, there is a 24-hour grace period in which certain things can be done. But the travel agent may have no such rule, and (even in the case of a published fare) might have to apply the standard rules from the moment of booking. In the case of a published fare, that involves paying the change fee associated with the current fare, together with the fare difference up to the new fare that you want. Again, this is something that you might usefully talk to the travel agent about in advance.