Getting from LHR to Harwich to catch the boat involved 2 transportation options.
The cheaper way is to go from Heathrow to Paddington station via Heathrow express. Then the tube or taxi to Liverpool St train station whereby you take a 1 1/2 hr train to Harwich.
Here's what they don't tell you.
I think in the end you may save yourself $35 vs the cruise ship transfer $104 US but I'm not convinced it is worth it because time wise it is very close -- 3 hrs.
There is a number of moving parts with the train option. You have to get ticket and lug luggage to the airport train. Then transfer to the tube during rush hour with regular commuters. Once you get to Liverpool station, you have to buy your ticket which is easy. But the train leaves once an hour.
It's complicated to try and calculate transport time. And if you were in line with me, there's 1 1/2 hr just to clear passport control. If you just missed the hourly train assuming everything else went easy, then you have time for another tea or coffee.
I took the bus transfer offered by the cruise company after researching the logistics. I had time but I didn't want all the moving parts after a long day and night of travel. So I slept on the bus.
I was tempted to take the train on the way back as I was staying in London until I read about the schedule tube works which was closing some key stations in the circle and district lines. My hotel was the South Kensington/Knightsbridge area. So again, I took the cruise line transfer, this time dropping me off at Victoria Bus Station.
Because I couldn't get to Paddington to take the Heathrow express, I took the longer Piccadilly line back to the airport. A pleasant ride at 6:30 am.