I live in London, so transport (rather than tourism) is my speciality here....
You can get from Heathrow to central London by:
(i) Tube (£4-ish) multiple stops, to South Kensington and then a short cab journey
(ii) Heathrow Express (currently £14.50), 15 minutes to Paddington and then get a cab or a tube and a cab (though getting a tube here defeats the object of not getting it in the first place!).
(iii) National Express coach to Victoria (£4). Takes 50 mins to an hour, then a
very short cab journey. Even if the traffic gets clogged up, you will still be on a comfy coach with air conditioning... something to consider over the Tube if you are here in summer and it is hot.
One thing to remember about the Tube (London Underground) is that access to most platforms involves some stairs. There are plenty of escalators for most of the ups and downs, but you are likely to meet stairs at some point. If you look at
Home | Transport for London they will have various maps that you can look at. The Tube map will show some stations with a wheelchair symbol - these offer platform access with no stairs (e.g. they have lifts) - unfortunately, these stations are rarely the ones you want them to be. One of these days I will count the stairs at the major stations (whilst hubby carries my bags!) and write a practical guide book to the Tube.
I cannot recommend strongly enough getting an Oyster card if you are stopping for a few days and plan to make use of London Transport - again, look at the London Transport site for more info. They are not complicated to use, particulary if you are just travelling within central London - just watch what other people do with their card, and off you go.
You get to Harwich by train from Liverpool Street Station. There are direct trains, and they often run extra services to co-incide with cruise departures. It takes about an hour and 25 minutes and (at the moment) costs £23 single for an adult after 9.30am. Group discounts are available if there are 4 or more of you travelling together (e.g. 4 travel for the price of 2).
National Rail Enquiries - Official source for rail information, UK train times and timetables for more details.
To get from your hotel to Liverpool Street, you can either get a cab, or get the circle line on the Tube (the yellow line).
As for what to do.... it depends if you like art (walk along the Southbank, take in the Tate Modern then cross the Millennium footbridge to St Pauls), theatre, museums (I love the Victoria and Albert museum - the fish slice collection is something to see), shopping (Harrods foodhall, Oxford Street - but go early), monuments & houses... (don't forget Buckingham Palace). A bus tour is easy on the feet, and going up something is always recommended for a good view of the diverse architecture - as is the boat tour. Finally, pop into a food store eg. Tesco / Marks and Spencers, grab some food for a picnic and take in one of the parks. Maybe walk from Piccadilly Circus (lights) down towards Green Park (perhaps book tea at the Ritz) pop into Fortnum and Masons foodhall (for a window shop) before heading through the park to Buckingham Palace before walking back up the Mall to Trafalgar Square then head down to Westminster to the Houses of Parliament, making sure that you pass Horseguards en route ... Is that enough walking for you????
J.