Well, we're assuming they somehow took care of the fleet. Taking over Ireland would be much easier than Britain itself, and would allow you to esentially starve them of US support and thus make your future invasion that much easier.
Operation Green called for the involvement of over 50,000 German troops. This outnumbered the Irish Army and would have been better equipped,better led,better trained and have a better doctrine. Ireland during World War Two was woefully underprepared. Ireland had a army and reserves were 13,000 reduced from 18,750 following the phoney war. At the wars outbreak, most were poorly trained.
Out of 298 required anti-tank rifles we had 4, out of 32 anti tank guns we had none, out of a required 924 bren guns we had 82, we had 4 of our 72 searchlights. An additional 4 medium AA guns, 4 bofors guns and another additional 4 searchlights were acquired after the outbreak. Small arms ammunition were around half of the requirement and rounds for mortars, anti-aircraft and field artillery were 8% of what was required.
Our navy was not worth a damn. Two converted fishing trawlers whose military capabilities were worse than merchantmen and six unreliable and awkward PT boats. Our airforce at its height in the later part of the war was 43 planes of a various assortment. Our situation was abysmal.
The greatest enemies of the Germans would face on the ground was the terrain, the weather (rain) and the poor road system. The Germans would enjoy quick and easy success in the opening days of such an invasion. The only serious resistance the Germans would receive would come from the North from British units stationed in Northern Irelnad. Planning for the defence of Ireland called for two British Divisions, 53rd Division and 61st Division to defend the Irish shores. RAF units have probably been brought over. The British made extensive plans to supply their troops. This is where the German invasion becomes problematic. The Germans now having advanced inland after being delayed by the poor transport system and Ireland's mountainous coast (Ireland is kind of like a bowl). They are now coming up against the much tougher and better supplied British troops. This would bog down any German any further advance into the Irish territory. The only possible problems with this are political. There is still anti-British resentment from the War of Independence 1919-1921 which could lead to resistance to the presence of British troops on Irish soil. There is also the possibility of an IRA coup to seize power and welcome the German's but I don't think so as the these men were opposed by the majority of Ireland in the Civil War 1922-1923. Also if such a coup took place and the military joined them, they're in an even worse boat than the German's. They are more likely to be a hindrance than a help. Taking the German's valuable supplies or simply being poor allies like the Italians.
Now, the miracle that allowed this German army to land in Ireland without being stopped by the Royal Navy has worn off and they will now proceed to blow the bejeesus out of any German ship in the waters surrounding Ireland and Britain. This would probably cut German resupply and reinforcement to a trickle if not halt it entirely. This German army is now present in a country that could not provide the means for an army almost one fifth of its size. It has no domestic arms industry, so the Germans can't press gang the factories to produce supplies for them as there is not even equipment or raw materials which could possibly be used to replace the German equipment and munitions. There will probably an abundant source of food so they won't starve at least. Now the German's only hope is British supplies which will probably be well protected as the British know how valuable as it is the German's only viable option other than surrender. The German's will probably not capture the supplies.
The German army now having expended the majority of its munitions, has two options surrender or retreat one is viable, the other is not. Surrender is their viable option. Retreat is an option but they have nowhere to go. Back to the coast, where then? Home to Germany. Nope the Royal navy is in the way. Grab the furniture and trees to build makeshift rafts and float to friendly land and be too numerous for the navy to destroy. Not many will make it.
So what have the Germans accomplished. They have lost 50,000 men and their equipment gallivanting about in Ireland. They have handed the ports of Ireland neatly into allied hands. This will help Allied ASW and offer greater security to Allied convoys in the trans Atlantic journey. It also opens makes Irish agriculture more open to the Allies, lessening food shortages and raising public morale. I'm sorry but I don't see anyway the German's really benefit from this.
You forget that six counties in the North belong to England and they have military bases too. The British are not unaware of Ireland and they have viewed Ireland as the backdoor to the England for the past few hundred years. This is not exactly unexpected and has always been a British fear.
I'm very sorry if this came across as too harsh but there are very wild things people say ''what if'' and ''but'' to. I simply drive home the point that it was in no way feasible or possible outside of a game.