Sorry to throw a fly into the ointment, but have you considered the option of getting a shuttle bus to Hatton Cross, followed by a 285 bus to Feltham rail station (5 minutes), which has fast trains to London Waterloo, a 10-15 minute walk from Aldwych? This would be quicker than the tube, but maybe slightly slower than the HEx option.

Also, as an additional Gatwick option, try the Thameslink service to City Thameslink, from where a 5 minute bus or taxi ride will get you near Somerset House.

For the rail options see southwest trains or national rail