January - London Falls
By January 8, OKW has reinforced by landing 11th Army at Bristol. This army will be responsible for the final push to Plymouth to eliminate all resistance in the Cornwall area. Meanwhile 15th and 1st Armies have done what they can in a broad-front advance, with surprising success:

Birmingham is taken, and 1st Army now sits on the western outskirts of the London metro area. The prize for 15th Army is Liverpool, but early attempts to attack the city have been very costly.
On January 24th, forward elements of 1st Army have managed to capture central London and desperately hold against a furious counterattack:

Manstein's 1st Panzer Group has finally been freed to execute its penetration to the northeast and has reached the beaches. The British have done exactly what we want here and shifted the bulk of their forces to the north away from Liverpool to meet the panzers head on, but 15th Army by itself is not able to do much more than hold the line. Reinforcements are on the way to kickstart the push northward.
The marine divisions from the failed landings at Dover have been brought in by way of Bristol and reinforce 10th Army as it continues to struggle to reduce the defenses at Plymouth.