The big shark in this scenario is undoubtedly King Phillip III of Spain. If he fully committed his army to helping Ireland, it would have all been over for the English.
Spain did send troops to Ireland's aid during the Nine Years' War. They were at the Battle of Kinsale, but there weren't many of them. O'Neill had been counting upon Spanish support, so this lack-luster response was a great disappointment to him.
Two treaties with England hung over Ireland and Spain in 1606. The first was one signed in 1603 by O'Neill, which committed Ireland to never ask for foreign assistance in its defense again. I can't see that being a problem. The Flight of the Earls in 1607 was all about O'Neill and Tyrconnell going to ask in Catholic Europe for foreign assistance in Ireland's defense.
The second had been signed by King Phillip in 1604. That guaranteed peace between Spain and England, ending the Anglo-Spanish War. Again that would only hold for as long as it served Spain for it to do so. It was thrown out in reality in 1625, as the Thirty Years' War kicked off.
So let's assume that neither Hugh O'Neill nor Philip of Spain saw either treaty as a long term barrier. But would that have meant that Spain would intervene?
It's a hard one to call. In ordinary circumstances, no, Spain would not and did not come again. But these are altered circumstances and the temptation may have been too much for Philip to resist. After all, Ireland would be a prize which could be used as a base against both the English and the Dutch. Very useful in the Thirty Years' War; not to mention great for hitting at British pirates, before they could sail into the Americas and attack there instead.
However, he would have to commit hard, sending far more troops than he did for the Battle of Kinsale; and he'd also have to agree to let O'Neill decide strategy. The Gaelic armies were not suited to the open battlefield tactics favored by the Spanish.
Nevertheless, it's infinitely possible in 1606. But not if O'Neill waited until 1607 to stage his rebellion, as Philip's financial situation was too transparently dire by then.
The Vatican would not have openly spoken out about war in Ireland, for fear of further repercussions against the English Catholics. But Pope Paul V would possibly be putting pressure on the Catholic monarchs in Europe to help the Irish, if such a massive anti-Catholic backlash had occurred in Britain.
Highland Scotland was another Catholic stronghold, and the closest of them all. Gallowglass mercenaries had already been in Ulster for years, fighting on behalf of their Irish brethren. That would have continued now.
But would the Scottish clans have been officially called for this fight? I highly doubt it. King Charles I was Scottish. The clans wouldn't have fought against him, even if the Irish Gaelic way of life was in jeopardy. Their loyalties were clear from the outset and would continue to be, in reality, in the following century and a half.
It was support of a dispossessed Scottish king which led to Culloden.
Which leaves France. This one is a little more complicated, in that it all rests on the domestic and religious pressures placed upon Henry IV of Navarre. He had converted to Catholicism in order to rule France; and had signed an edict allowing religious tolerance in his country.
France was notable for its pacifistic stance at this time, not to mention being busy founding colonies in what would become Canada. I don't think that Henry IV would have committed his navy for Ireland. There's no precedent and little that would have changed to force that situation now.
It really did all boil down to King Philip III of Spain. On balance, I think that he may have helped in 1606, in which case, Ireland most certainly would have won. Until next time.