How does succession work in a tanistry empire

King can never be a vassal of another king, and there is no “partial vassal” or something like that (and the same applies to all the other levels too).

The duchy will become part of your kingdom (de facto, this won't change which kingdom it is part of de jure). But if the crown authority in the other kingdom is high enough, you won't get the duchy (it prevents titles to pass out of the realm through inheritance).

Succession laws for multiple kingdoms (and no emperor title)
(For your example where you are the King of Ireland and Wales)

1. King of Ireland and your lower titles that are de jure part of Ireland are governed by the succession laws for "King of Ireland".
2. King of Wales and your lower titles that are de jure part of Wales are governed by the succession laws for "King of Wales".
3. [Duke and lower] titles that are not de jure part of any of your King titles are governed by the succession laws for your primary title

Changing a succession law
If you changed the succession law for King of Ireland, and King of Ireland is your primary title, all of your de jure Irish titles and titles outside any of your de jure kingdoms (#1 and #3 above) will be governed by the new succession law.

You would need to change King of Wales separately. (In the law interface, click on the coat of arms for Wales.)

Succession laws for titles held by vassals
Your vassals have their own separate laws for succession which could be totally different from yours. For example, you have have agnatic primogeniture as your kingdom succession law, and your vassal duke could have agnatic-cognatic gavelkind as his succession law.