Why are so many grumpy people on this thread?!
I have such happy memories of days out like this with my grandparents. You’re going to have a great time together and it’ll be a lovely memory for him.
OP, you could park in the centre but you’d still end up on public transport between the different chosen locations, so you may as well spare yourself the extra drive and park further out.
If you’re parking in Richmond as you mentioned, it might work well to do things in this order:
District line from Richmond to Victoria, 10 minutes walk to Buckingham Palace.
Walk through St James Park (really lovely park with interesting birds on the pond) to Westminster where you’ll see Big Ben.
Jubilee line from Westminster to Baker Street. Madame Tussaud’s is literally next to station.
Bakerloo line (or bus) from Baker Street to Oxford Circus, 5 minutes walk to Hamleys.
15 minutes walk or a couple of stops on bus to Lego store. There is an M&M’s place next to it which kids seem to love.
From there, you are 15 minutes walk, via Trafalgar Square, to Embankment station where you can go back on the District Line to Richmond.