Well well well! I found there is this site where they leet volunteers to go and have many elderly homes in London!!! Here are where they listed they have it:

Town/Borough
GREATER LOND0N -

Barking and Dagenham x 2
Barnet
Bexley x 2
Bromley x 5
Camden x 3
Croydon x 2
Ealing x 3
Enfield
Greenwich x 3
Hackney x 2
Hammersmith and Fulham x 2
Haringey x 2
Harrow x 4
Havering x 2
Hillingdon
Islington
Kensington and Chelsea
Kingston upon Thames x 3
Lambeth
Lewisham
Merton
Newham & Tower Hamlets
Redbridge
Richmond upon Thames
Southwark x 2
Waltham Forest x 5
Wandsworth x 2
Westminster x 6

The nearest one to Paddington is these few:
Kensington and Chelsea
Lambeth
Islington

The webstite is - http://www.contact-the-elderly.org.uk/page.php?page=locations

I hope this will work out! :)