I just bought some tickets, and when I checked for the change in plane
in London, I forgot to check whether the change in plane takes place
at the same airport.

Now that I've bought the tickets, and found that I arrive at Heathrow
and must depart at Gatwick, how do I get us (me and Mom) over from
Heathrow to Gatwick?

More details: the plane is supposed to arrive at Heathrow a little
before 11 a.m. GMT, and is supposed to depart Gatwick at 1:45 p.m.
GMT.

This raises issues like, how long will it take to clear customs
post-9/11 and Richard Reid (the Shoe Bomber)?

Assuming we clear customs quickly, what is the fastest and most
convenient way to get to Gatwick and somehow manage to grab some
lunch? Is there a shuttle between airports? Will we have to cab it?
Whatever ideas come to mind, please do not forget to include
approximate cost for the transport.

The fastest would probably be to take the Heathrow Express to Paddington,
tube to Victoria, then Gatwick Express to atwick. This will also be very
expensive. o cut costs you could take the tube to London.