Pre-booked London in 7 days followed by Heart of Ireland for 7 days. What’s the best way to finish in London and get to Ireland for tour starting the next day?
London (any airport) to Dublin is one of the most freqent routes in the world. Sadly it's not a lot of use to you because the Heart of Ireland tour starts in Ennis and you need a flight to Shannon, which only has a couple of flights per day. Start looking early, there are Aer Lingus from Heathrow, the mid morning flight is not too expensive if booked early.
Ryanair fly from both Gatwick and Stansted to Shannon, just be sure you check carefully when choosing luggage options etc. They will also try to make you pay extra to select seats- it's a very short flight so if you're not bothered about where you are sitting make sure you don't choose that option.