Flying through AA on British business class (club exec).

Several ?'s:
Are we able to access all lounges at LHR?
Does this include the Elemis Spa?
Is Club World the same as Club Exec?
For showers, do they provide shower shoes?

Lastly, we arrive 9:30am @LHR and depart @ 7pm from LGW. From what I read, we need to figure at least 3 hours. Would you recommend hanging out at LHR for a few hours before transferring, heading straight to LGW or is there any time for a side trip?

Thanks