Janice, The Ardreys came originally from Scotland (see my note in the Gen. For. re: Ard-airigh). I believe the Ardreys were part of the protestant plantation of Ireland in the late 1600s. A 'William' arrived in Virginia, USA in 1665. It may be that about this time other Ardreys moved to Ireland - this would make sense.
The Ardreys were, I believe, from Mullnahunch, a very small place, near Donaghmore. They were registered and baptised in Donaghmore.
As for your Jane. I believe her ascendants may be as follows, although this is speculative to a significant extent. She had one sibling I know of - the Robert you refer to. So Jane - father Wm. b. 1779 - gfather Wm. b. 1743 (Co Tyrone) - ggfather Wm. b. 1714 (Co. T.).