I stayed at the Heathrow Marriott on Thursday evening for the first time. The shuttle service is provided by "Hotel Hoppa" buses. The specific one for the Marriott Heathrow is H3. The service is GBP 3.00 each way. The service from LHR to the hotel was very slow, waited about 25 minutes, but the return to LHR was excellent, right on time. The "Hotel Hoppa" stops at all terminals so your specific airline / terminal is not a problem. The Marriott itself was fine. A newer property with good size rooms (by European standards). Food was good as well.