After a bit of searching here and elsewhere, it seems that T2 is recently rebuilt (I have not been through LHR in about 5 or 6 years), so I'm having trouble finding relevant threads that talk about the current state of affairs...

I am departing on a morning United Flight to LAX and wondering which of the nearby Hilton properties involve the least complex logistics (I'd rather have a 15 minute straight walk than 20 minutes & three different buses). I'll be staying at the Hilton London Tower Bridge for several days prior, but will transit to the airport in the evening to avoid the long taxi ride in the morning.

Hilton London Heathrow ("T4")
Doubletree Heathrow
Hilton London Heathrow ("T5")

Thanks in advance for your help!