Somerset House

Somerset house has a privileged location with The Strand on one side and the Thames on the other - when it's warm enough you can sit out on the terrace overlooking the river (although the road is in between!)
They have various temporary exhibitions so check the site: www.somersethouse.org.uk and in winter they have the famous outside all weather skating rink sponsored by Tiffany & Co. The skating rink has a special area for children where they can skate with an adult and special supports in the shape of penguins - very cute. In addition there are various activities for children and teenagers so again, check the site.
There are plenty of lifts between floors once you are in, but do check on a map or call ahead to check which entrances are accessible as some of them have steps.
There are changing facilities on the ground floor by the deli (in the disabled loo) but there is also a dedicated changing room with a sink, changing table and tissues next to the lockers (which are also handy!) on the LG (the Embankment entrance).
Unfortunately none of the cafes/restaurants provide high chairs, but the deli in particular has lots of space around the seating, and you can also take your teas etc through into the Seamen's Hall where there are comfy sofas and coffee tables.