Unfortunately, as you're coming from Atlanta (or even if Orlando was convenient for you) all direct flights from those places go to Gatwick only, not Heathrow. However, this is not that much of a problem and certainly better than wasting time connecting via some intermediate point.

There is a direct train from the Gatwick Airport train station (underneath the main, South terminal) to Reading, at least every hour, and you can connect there on to Cardiff, again at least every hour. You buy a ticket from Gatwick Airport station right through to Cardiff. No need to step off the railway platforms on the entire journey.

If you decide to stop over in Central London in one direction or the other, the Cardiff trains use Paddington station while the very frequent service out to Gatwick airport operates out of Victoria station.