If you don't check bags, as soon as you're off the plane follow signs for Flight Connections. A bus will then take you airside to T4 (so you don't enter the UK). You then clear security at T4 before proceeding to the gate.

Check in online if you can to avoid waiting for boarding passes at the transfer desk.

BA will not usually allow you to standby for an earlier flight on a non flexible fare, although I think most - if not all - BA European fares are now changeable for a fee.

Depending on the cost you may want to just look at rebooking on Air France as they also use T4.

Also bear in mind that BA tend to be a little more conservative than US airlines in their view of what is acceptable as carry-on baggage.

Probably too late now anyway, but booking your return from the UK as opposed to France also means you'll be paying out an extra $300-$400 in departure taxes between the 4 of you as well...

As I think you'll have figured out, you're far better off doing the research before making the bookings.