I have this in my "trip plotting" document and have no idea where I picked the info up from (the boards, I suspect!) Hoep its of use to you:
"From London Liverpool Street train station take the "Cruise Train" from Liverpool Street to Harwich International station. On your day of sailing, there should be (at least) one train going direct to Harwich International station. Current timetables show a daily train leaving Liverpool Street at 12.25 and arriving Harwich International at 13.38. Good websites to look at for travelling to Harwich are www.harwich-intl-port.co.uk/hip/ and www.ger.co.uk (this is the website for Great Eastern Trains which runs the "Cruise Train" service). If you don't find what you need on the website, Great Eastern Trains Customer Services on (UK) 08459 505000 can help. The standard single fare is £17.70."
I've also heard that if you buy a return ticket on the actual train its only £20 in all.
Haven't tried it since this is my first trip to Europe.
I agree with Anna-Maria. We are also on the 05-10 sailing of Constellation, traveling from GA. We are flying into Manchester a few days early. On Saturday morning we will leave from London to Harwich by train.