Frank, Re-read Nigel's latest posting. This is excellent advice. Also look at this tube map from the Transport For London (tfl) website. Click on the tab that says "Large print map":
You may want to print this map if you have a color printer. At any rate, study the map and tube lines. Follow the Piccadilly Line (dark blue color) that Nigel mentioned around the map. Notice the names of the stops it makes. These are the areas that Nigel mentioned in his last posting. These are all good areas for your hotel.
If you want to stay in Bloomsbury, I can recommend the following hotels.
Hotel Russell; take the tube line Nigel mentioned (Piccadilly) to Russell Square. Exit the tube station, and you are steps away from the hotel:
Walking distance from the same tube station is the Radisson Blu Edwardian:
I have stayed at both of these hotels and liked them. The Radisson Blu is the more modern of the two.
The neighborhood of Bloomsbury is also the location of the excellent British Museum.
If you are wanting to be near Paddington Station, you will have other tube lines accessible from there. There is a Hilton hotel connected to Paddington Station by a footbridge. It offers the convenience of going directly into your hotel if you arrive at Paddington from Heathrow.
When you wrote, "We thought we would figure out the best way to leave the Heathrow Airport and then try and find a hotel that would be easy to get to with our luggage in tow." This hotel fills that requirement. I have stayed at this Hilton twice and liked it very much. It is not "charming", but it is convenient and has comfy beds.
Keep in mind that both the Heathrow Express and Heathrow Connect trains run from Heathrow Airport into Paddington Station. Heathrow Connect is the less expensive option, and the one I would take.
Keep in mind what Nigel said, above, that your train to Cardiff leaves from Paddington Station.
Katy also gave you excellent advice. Her recommendation of the car service is a good one, especially if you eventually choose a hotel that is a pain to get to, and with lots of luggage.