Plenty of people take the local buses with child(ren) and large luggage. But understandable if you don't want to.

The Heathrow Express train goes non-stop from Paddington to T2/3 then T5, every 15 minutes.

The TfL rail train has 5 stops between Paddington and T2/3 then goes to T4, thus it is a bit slower, and currently only runs every 30 minutes. It is less than half the price of the Heathrow Express at walk-up prices.

About 3-5 minutes walk including escalators or lifts from the relevant station to T3, T4 and T5. T2 is about 10 minutes walk plus lifts. The walk to T2 is exposed to wind around the terminal entrance, T3 slightly exposed, T5 is sort of indoors and T4 is completely indoors