I'd skip the Shard and London Eye, they're on the South Bank and you'll eat up time getting there and back. Also, your travel date is nearly the shortest day of the year, so you'll be losing daylight quite early.

After the Palace and Westminster, get on the District or Circle Tube line at Westminster to the Tower Hill stop for the Tower and Tower Bridge (both adjacent to the tube stop). Back to Victoria to pick up your bags, north on Victoria Line to Green Park to pick up the Piccadilly Line to Heathrow. Figure 45-60 minutes to get there on the Piccadilly. If you're flying long-haul out of Heathrow, get there at least two hours before departure. Even though it's a Saturday, it's the week before Christmas and there will be a lot of traffic. Give yourself plenty of time to clear security. If you're flying BA out of Terminal 5, add some time to get to the gates at 5B or 5C, if necessary.