You can take the airport baggage carts to the end of the car park area, there are then the usual knee height steel hoop barriers at the entrance to the passageway over to HGI. At that doorway there were several standard hotel baggage trolleys. So you'd have to transfer the bags from HAL's trolley to the hotel trolley (which I guess you would have to do at some point anyway, I can't see them allowing HAL's trolleys to go to the room). I've amended post 55 to include a photo showing this.

That apart, it's step free for trolleys for both T2 and T3 Arrivals and Departures using the lift/elevator routes. T3 is approximately 650 metres from the hotel. For T4 and T5 they don't allow trolleys on the platforms of the rail or Underground stations for obvious safety reasons, so you would need to carry bags a few metres at the rail stations, and slightly further at the Underground stations. There are then trolleys available at the arriving stations.