From the 1942.2 rulebook: “An island or island group is a single territory surrounded entirely by one or more sea zones. A sea zone can contain at most one group of islands, which is considered one territory. It is not possible to split up land-based units so that they are on different islands in the same group.” The illustration for the rule shows the Japanese home island group, which consists of one large island labeled Japan and three smaller islands with no names. According to the rule, these must be treated as a single territory, regardless of the fact that they are separated by water.
In 1942, the large territory labeled United Kingdon and the unnamed territory north of Eire have the same relationship as the large island labeled Japan and three smaller islands with no names: they must be treated as a single territory, regardless of the fact that they are separated by water. It’s unrealistic, but that’s what the rules say. So yes, you could invade London without transports if you had land units on Eire – but remember that the “without transports” concept in this case is not a pure no-transport situation because you would have needed transports to get them to Eire in the first place.
A similar example from the 1940.2 rulebook: “Central America, containing the Panama Canal, is one territory, so
no land movement is required to cross the canal within Central America.”
In 1940, the territory that “Northern Ireland” is considered to be a part of is Scotland, not the part of the UK in which London is situated. So in 1940, an Eire-to-London invasion would be via Scotland, not a direct invasion.