Dream trip? Book a car pickup from LHR (see the thread in this forum for recommendations of companies to use)
Everyday travel? Take the tube. It's got plenty of room for luggage and starts at Heathrow so will be empty when you board, and is very easy to board with luggage. It takes an hour and is somewhat boring but workable.
HEX? Pointless waste of money with extra hassle transferring to tube or taxi at Paddington to get to St Pancras for a scarely-faster journey.