Bacause of your Class of Travel you should have time to do your own Transfer with Luggage, assuming of course you have "Right of Entry in the UK"

1] At Check-in in USA ask for bags to be Tagged only to LHR -This could be refused if your fare does not allow it

2] Fastrack Immigration at T1 or UK/EU

3] Collect Priority Bags at T1 [Hopefully off First]

4]Exit Green Channel

5]Take Bags on Cart down to Heathrow Express

6]Transfer on HEX from T1 to T4

7]Re-check Bags at T4 Club Desk

8] Fastrack security in T4 Departures

If however you dont have right of access into UK or Bags can only be checked directly through to Final Desination change to a Later Flight LHR-GVA..relax and enjoy the Lounge Facilities at T4 - Gate One.