Spanish (or French, come to think of it) intervention and support would be vital for an independent Ireland to be established during the Elizabethan era- I'm not certain that Hugh O'Neill or any other Irish lord could successfully rally all of Ireland to rebellion without some sort of divine intervention, what with clan politics at the time be painfully complicated. And, even if O'Neill did manage to rally everyone to the cause, Bess' advisers would probably either push for just invading Ireland again and again until rebel leaders give in or offering concessions and then breaking promises once peace is made.
Really, the Irish would've had a much better chance if they weren't divided between so many squabbling clans. And didn't have English lords breathing down their necks already. And had fair political influence outside of the isles.