NP ... your journey will be as seamless as UK can manage (E&OE)
LGW - Premium Gatwick Passport Control, £9.99 each.
NEX - Flex ticket in case of Early/Late arrival, baggage handled by coach driver.
LHR - Drop-off at Bus Stop 13, below the Club desks in Zone H [a short flat walk to the elevators]. Priority Security as Club Europe Pax + Club Lounge Access.
Only downside will be the on-board experience in CE to TLS
The annoyance of the LGW-LHR connection [or the other way round] is that you really have to start over from square one at the next airport. There is no "connection" for either pax or bags.