There's a bus company called Diamond that run mini-bus services from towns close to LHR, including service N555, it's mainly for HAL staff, who subsidise the service, but the public can use them. Contactless payment is accepted. There is a bus due from LHR T4 bus stop 10 at 04:30 and this gets into the Central Bus station, by T2, for 04:48. If that's a bit early then you may as well wait until 05:00 when the tube starts from T4. Otherwise get the hotel to book a private hire, which is going to be in the £30 area. If they have Hilton points spare, then HGI T2 is possibly an idea since the redemptions aren't always at silly amounts.