OK, this is only for vanilla, but it should work.
1) neutralize the RN. Your AF will be very useful in this regard, as in 41 your navy is probably worthless. use small surface fleets to draw the RN out, then pound them with naval strike with TAC and CAS. when they retreat to port, blow them up with port strike. This process might take a while, but it will be worth the effort when you try and cross the channel.
2) The landing prep. Use your AF to conduct log strike on your chosen strike area. wait till infra is reduced to as low as it will go. You might also throw some aircraft on interdiction / ground attack once infra is well reduced, but only against small stacks. Also, after infra is reduced, use installation strike to remove fortifications from the area.
3) The landing. A gamey way to attack is to launch some paras into Cardiff, which the AI often doesn't defend as there isn't a beach, then simply sea transport troops to reinforce. If you don't want to do that, you're going to need marines, especially if they have lots of defenders. Set up a couple invasion waves, and set it up so that you have 3 (max for sea invasions until MUCH later) divisions attacking per wave. Then as one wave gives out, send in the other. You're also going to want the AF on interdiction during this time, as well as any surface ships (your starting BCs and CAs especially) on shore bombardment. Have a lot (at least 6-9 divisions) in transports ready to reinforcethe beach head once seized.
4) lodgment. After you get the message that the enemy has been defeated, try shifiting your AF to ground attack to destroy the retreating enemy. If this doesn't work, and they attack other forces, don't worry, just shift them to interdiction, that's useful too. As soon as the invaders seize the province, land those reinforcements. I also choose to take this time to drop paras (2 corps / 6 divisions) behind the lines to cut enemy supply lines and interdict reinforcements. At least one corps should be dropped around Edinburough to prevent the island gambit that the AI often uses.
5) exploitation. I assuming you know how to fight once across the water. Remember to prioritize mobile units and HQs across the channel.