If there is a type certificate holder the aircraft will be Annex 1. If or when that company closes, sometimes years after the authorities can decide to move it to annex II, as has happened with Austers etc. But remember that the CAA was for probably more than 15 years the de facto type certificate holder. The critical point is the owner does not get to choose. He/she can make representations but nothing more.
Virtually all PA18/150s are Annex 1, the smaller older aircraft can be Annex II, depending on the authority. Homebuilt replica Pa18s can be Annex II.
A considerable number of deHavilland's Austers etc have moved from I to II - but only after the authority ruled they could and the LAA agreed to administer them.
In places like France the situation is at least as complicated with for instance Gardans moving from Annex I to a restricted C of A category, still administered by DGAC