RCI have coach transfers from LGW to Harwich.
Its not easy by train with that amount of luggage as you would have to change through London. I was on Jewel of the Seas 2 years ago and there were dedicated coaches from Heathrow direct to the ship which is the best way to go. There maybe also some from Gatwick. Suggest you check with R.
Hello....Carrying that much luggage around is not going to be a brilliant start to your holiday. Check to see if there is a coach arranged for the trip as Harwich is really awkward to get to. If no, look up your particular cruise on the boards of another web site which has both cruise and critic in its name then ask if someone would like to share a taxi with you. Arrange one from Harwich Taxis, not at the airport, to meet you at whichever airport, and here I would suggest Gatwick, and be taken right to the ship in comfort. Cost is about £120. It is possible to hire a car one way but if you have never driven on the left before that could also be an inappropriate way to start a holiday.....Heathe.