Dreams of Traveling

Looks like I’ve found the perfect group to ask about traveling from Southampton to London! My husband and I are returning to the Southampton cruise port via Royal Caribbean on a Friday at 6am. We have to catch a 1:55p flight at Heathrow same day. What’s the best way to get from the port to Heathrow? We do plan to carry our luggage off the boat, no problem there. Train or bus? Anything direct? Thank you in advance for your suggestions!