I think it's easier to do London first. You'll have time to explore and get over jet lag.
Then fly to Shannon. Forget this fly to Dublin nonsense. From there it's just over a three hour train ride with a change of trains in Limerick. And don't forget you still have to get from the Dublun Airport to the train station in town. That's 45 minutes to an hour. From Shannon Airport to Ennis it's about 30 minutes. Both the train from Limerick and the bus from Shannon Airport let's you off at the same place.
Get an open jaw ticket London to Shannon, Dublin to London. While you do go through immigration at Shannon it's easier than Dublin. (For those about to jump in to say it's an open border--U.S Passport holders have to go through immigration in Ireland.) However, there is no immigration in the U.K for arrivals from Ireland. Even us Yanks. I have traveled both ways many times.