Irish Invasion..
I share a lot of the scepticism of others. I tend to the view that if German forces had successfully landed and conquered Britain, Eire would have had little option but to sign up to the Axis. If Britain under Halifax had signed a peace treaty with Germany, Eire would have no say in the matter.
The only even remote possibility for a successful German invasion (apart from after Britain has fallen) would be perhaps in April. The successful invasion of Norway starting April 9th 1940 was as much coup de main as landing by force. Now, Eire has a similar population to Norway. Could the Germans, by adroit use of merchant shipping, successfully occupy key ports such as Cork or even parts of Dublin and then land paratroopers to bolster the invasion (this would happen INSTEAD of a Scandinavian campaign rather than as well as) ?
My guess is the Irish Army would be forced into the rural hinterland and, with air support, the Germans would soon be in control. As for the British, remember the BEF is in France at this time with the best of the RAF units also stationed in France. As happened in Norway, the British and French would have insufficient time to react before the Germans had taken control of Eire.
If this happened in April, it would force the British to redeploy units to Ulster and western England and would increase the fear of invasion in the country. With the BEF depleted, the remaining Anglo-French forces cannot even withdraw to Dunkirk and are lost leaving Britain surrounded and weakened with Luftwaffe units able to bomb the west and north of England and drawing key RAF squadrons away from the south and east in the critical summer months.