The Irish militia, as I recall (it's been about ten years since I studied it in any detail), wasn't really sent to Britain, not even after the union. I think there were proposals to do it, based on the idea that getting the Irish militia further away from sources of local sedition would be a good thing, as well as getting them trained by British officers on British soil. I don't think it actually happened though. The British militia did go over in 1798 but I *think* (I may be wrong) Ireland was mainly garrisoned by soldiers. This of course was another argument for establishing an interchangeable militia system between the two islands, as it would have freed up more regular forces, but no go.
The government did try repeatedly to formalise the process of "drafting" militia into the regulars from the 1790s onwards -- the biggest draft occurring just prior to the Helder campaign -- but repeatedly ran into trouble with the county magnates, most of whom were highly influential in Parliament, and on whom Pitt particularly was strongly dependent (witness the trouble he had in 1797 when a third party interest formed under the leadership of Sir John Sinclair). I recall quoting a letter from Lord Auckland referring to the militia question having the same effect on the country backbenchers as music did on people bitten by a tarantula. In fact in 1805 Pitt passed off a proposal to move 17,000 men from the militia into the regulars as a militia reduction rather than a draft, just to avoid criticism! There was an entrenched idea, rather outdated, that the militia was an army that reflected the composition of the country: landed gentry commanded it, and the ranks were supposedly filled by an assortment of propertied men with a direct stake in society. The reality was rather different, partly thanks to the substitute system and partly because of the obstruction to recruitment into the regulars. I can't remember offhand when the militia draft became a regular thing -- I think it was probably established as part of the system accompanying Castlereagh's Local Militia Act, but you can probably tell me more about that than I know myself!
Checking the appendix of my thesis (listing the national defence acts from 1794 onwards), a militia draft was proposed (and failed) in 1798; there was the Helder draft in 1799; there was another draft in 1805, and a further draft from the Irish militia in 1806 (but the Irish militia, as I implied above, wasn't considered a very good defensive force anyway); then there were semi-regular drafts in 1807, 1808, 1811-14, which does indeed coincide with Castlereagh's Local Militia.