Do you ever play that historical game of trying to guess what would have happened if the thing that did happen hadn't happened. We can play that with the short appearance of Tuda in our story.
Tuda was, for a little while, the fourth bishop of Lindisfarne. So he followed Colman, who withdrew from the bishopric and returned to lona after the Synod of Whitby (664).
Bede tells us that Tuda had been educated among the southern Irish. So, with regard to the date of Easter and other problems discussed at Whitby, he would have absorbed the continental position, not that of lona. The southern Irish had already decided to accept the 'Roman' dating and the Roman position on other matters in dispute; this is why we can't see the Synod of Whitby as a contest between the 'Romans' and the Irish as a whole, but rather between the continentals and the northern Irish led by lona.
Bede tells us that Tuda was 'a good and devoted man'. He had been consecrated bishop in Ireland, but had come from Ireland to Northumbria while Colman was still bishop and there was occupied in teaching 'the true faith' i.e. the continental view. So he was on the spot and available when Colman left. We may assume that King Oswiu noticed him, for Oswiu was very much a 'hands-on' king in the affairs of the church. So he was appointed bishop of Lindisfarne, with the hope that he would be a reconciling influence after the Synod.
But cruel fate took a hand and Tuda almost immediately caught the plague and died. 664 was a bad year for plague, which we presume to have been the bubonic type, similar to the Black Death seven centuries later. Cedd, Bishop of the East Saxons, who was in Northumbria for the Synod, also caught the plague that year and died at his monastery at Lastingham.
The death of Tuda created a vacancy which was eventually filled by Wilfrid, who moved the bishopric to York, and left a certain amount of havoc after him. Wilfrid was a great man, but he was not a reconciler. So it fell to Eata and Cuthbert to heal the wounds at Lindisfarne and eventually to restore its position of leadership.