I would urge you not to pre-book a train ticket. You'll be tied to the booked train, so that means unless you time it to perfection, either hanging around waiting at Gatwick, or having to buy a new ticket because your train has already gone.

Given the lack of left luggage facilities in London, I think you have the right general idea.

Let's say you get away from Victoria left luggage at 10.30am. 10 mins walk to the palace, then another 10 minutes walk up to green park tube. 11am on the Jubilee line to Waterloo, will have you there by 11.15am.

Quick turn on the London Eye (pay extra for the quick entry) will take you to through to about 12.30pm.

Take the Thames Clipper boat down the river to Tower Bridge. Sit at the front for the best view. That should have you at The Tower of London by 1pm or so. It'll also let you grab a glimpse of places like St Paul's, The Globe Theatre etc.

That gives you a decent spell of time to see the Tower.

5pm board the tube at Tower Hill (District or Circle line) for Victoria

5.30pm be back at Victoria incase of delays in picking up your bag

6pm leave Victoria on the District Line, changing to the Piccadilly Line at Barons Court - just walk across the platform - for Heathrow and that would have you at the airport for 7pm or so.

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The one challenge I see here is the risk of pre-Christmas crowds. You're reasonably far away from the main shopping areas (apart from Green Park tube station) and it's a Sunday, but London does seen an influx of visitors in the weekends leading up to the year-end.