Hi and a very good morning to you,
There is a shuttle bus that operates from Central London, but starts out quite early in the morning, and seems to take up to 4 hours
so probably not the first option.
Trains run from London Liverpool Street into Harwich every hour on the hour and take 1h17m, but if you are sailing on a weekend, please recheck.
National Rail Enquiries - Journey Planner - Trains Times and Fares
The ticket's AREN'T cheap though
the best seem to be around £30 ($50) per person EACH way, so if you are in a group, a taxi transfer may be better as it will be door to door without any hassle of getting to/from the stations. If you put in a few enquiries you may even get to see a bit of the City from the cab as it makes it's way from the centre.
I'll keep thinking, and no doubt others more central to London will see this and conrtibute.
Wish you a wonderful day,
Alan & Katrina