I wonder if you couldn't get it done by beefing up the Luftwaffe. First, improve their regular combat aircraft so that they can actually defeat the RAF (earlier jets?). Second, change their strategy so that they go after the RAF planes and airfields instead of randomly bombing London. Third, have them develop several wings of aircraft specifically for anti-shipping. After the fall of France, with airbases right there on the Channel, the Luftwaffe could use land-based air to keep a broad lane of the Channel clear of the RN. Of course, the Reich would also have to build some real landing craft for the first wave of troops going across the Channel.
Basically, you'd have to have the following scenario:
the Luftwaffe inflicts a serious defeat on the RAF, either destroying them in place or forcing them to shift up to the north.
The Luftwaffe next clears a lane in the Channel with both mines and aircraft.
The invasion force then moves in through this cleared lane, landing on Britain, with the goal of capturing a port so that regular German ships can land reinforcements.
I'm not expert enough to say if this would work or not, but it seems as if you'd have to have almost everything go right for the Germans and nothing go right for the Brits, and the chances of that seem unlikely....