This was what we did in the end on Tuesday:
We left our hotel at about 8am, took a tube from Paddington station, got off at Bond Street. It was an experience taking a tube in the rush hour.
There was a toy store just out of Bond Street station exit, but it was still closed. So we decided to walk down to Grosvenor Square. It was drizzling, so we took a shelter at the 11 September Memorial Gardens at the east of the park. Was greeted by a robin, and was quite surprised it didn't seem to be scared of people, unlike where I live. It came close just by where we sat, and we fed the bird some crumbs. It then sang, perhaps to express its gratitude.
At about 930am we headed to the embassy and it was 1130am when we were done with the application.
I decided to follow @Oululainen's suggestion of visiting Hamleys. My son had a good time there. There were some toys to play with at the store, and he got what we wanted, so he was pleased. Had to break for lunch at about 1pm, which we had at a pub by the alley just behind the store. Went back to Hamleys after lunch; it was already 2pm. So we quickly made the purchases and proceeded by bus to Paddington Station to catch our train.
Realising that we would miss the 3pm train and would have to take the next one at 4pm, we stopped by at Semiramis to buy some ice cream for my son. The dondurma got nuts in it, and knowing that my son dislike nuts, I bought him the ordinary ice cream instead, which the shop also sells. He was happy.
We reached Paddington about 30mins early, so took a quick trip to the Paddington Bear store, where the Paddington Bear statue that @Lego Stormtroopr posted is now located.
That was our day on Tuesday. A word of caution to others visiting Hamleys: I think the shop assistants, many of whom were youngsters, were a bit annoying.