I'm afraid this is going to be extremely tight.

Early morning is a very busy time for LCY. LCY is bursting at the seams, and it is not uncommon for inbound flights to be delayed for no other reason than lack of space at the airport (my last flight to LCY was delayed by over half an hour)

If the flight is on time then - assuming your family already have their BPs - it's do-able. Using DLR and Tube you need to allow about 90 minutes. Don't know how long the taxi/HEX connection takes. Going by taxi all the way took me just over an hour at a slightly later time in the morning a few weeks ago. (I didn't pay for the taxi, I hasten to add - OS did after cancelling my flight from LCY to VIE)