Hi Dan,
Yes there should be luggage carts (or trolleys as they’re known in the UK) available in the reclaim hall.
At T4, National Express serves bus stops 13 & 14 outside the exit from the arrivals hall, on the same level. So a very short walk.
You should easily make the coach, but please note it departs T4 at 4:15 - make sure when booking you select Terminal 4, NOT Heathrow Central Bus Station as they are nowhere near each other.