I'm going to assume that the driving itself won't be too much of a challenge for you---the "other" side stuff. There are a lot of road rules, for example--in England, there are not a lot of stop signs on every corner, it can get a little confusing when everyone to your right has the right of way.
Parking is a HUGE challenge and expensive.
For the most part, especially within London (and I think also Dublin & Manchester) it's easy to use the public transport and go where you need to go. In addition to the ease and saving money, you will be experiencing real life instead of insulating yourself in a car. (I like to see what it's really like...)
If you're wanting to stay mostly in the city---don't bother with renting a car. If you want to explore very rural areas, then a car is definitely needed...perhaps you could just rent one for those particular days.
Plan on having a great time, whichever way you go!!
An American living in the UK