This early period is straightforward and the Mamelukes shouldn’t put up much of a fight. It is very important to protect your dockyard in Alexandria so station a couple of troops in it and begin constructing a naval fleet as early as possible. Make sure you leave a reasonable force in Alexandria with artillery because the British are quite likely to land a force there.
Having played the campaign a few times the British seemed to turn up randomly at various points in the campaign. It is very helpful to construct a few decoy ships, a few sloops perhaps, so you can send them out into the Mediterranean and track the British fleet. Do not engage them in naval battle though. Later on in the campaign you’ll want to use your decoys to keep the British fleet busy so you can sail an army across the water to attack the British in Nicosia but for now we’ll focus on securing Egypt.