Tilbury is not a great tourist stop, you may better to consider an airport or London hotel. Although I believe there have recently been vast improvements at Tilbury to promote the cruising terminal. The Heathrow airport hotels IMO are not great- other than the price- but given jet lag perhaps the best option.
If you want to go into London, the Tower Hill suggestion above sounds a reasonable mid-way point. The Tower Hotel there is right on the river too.
The recently opened Elizabeth Rail Line, in purple, offers step free access from Heathrow to London and beyond. See TFL (Transport for London) website for maps and connecting stations.
Amendment: Have just found this via Google
How to get from Tilbury cruise terminal to London?
The Tilbury cruise terminal is quite basic, housing a café, general store, and pay phones. As you exit the building, you’ll see a taxi rank which can take you into London. A much cheaper alternative is to take one of the shuttle buses which run back and forth from the terminal to St Pancras International Train Station in central London.