It's January. Daylight hours are far shorter. Don't count your arrival day or final day as part of your total # of days on the ground because you really aren't going to accomplish a whole lot of sightseeing on those days (if any). Does that still leave you with 14 days or 16 days or is it less? Is your plane ticket open jaw (flying into one city - like Shannon and then out of another city - such as Paris)? Or do you need to spend even more time circling back to your original point of entry because you have already booked a roundtrip ticket?
You're trying to squeeze 3 entire countries and a city in a 4th country into a pretty short time frame. You don't seem to have accounted for the hours you will need to spend packing up from one hotel, checking out, getting your train, plane, bus or rental car to the next destination, finding your new hotel, checking in, unpacking, etc. That's all time lost from your trip.
Check the local library and watch some DVDs or YouTube videos on the destinations you have in your head and start figuring out how you will get from one place to the next, how long it will take you and what you really must see in those places. This will give you an idea of how much time you might need to spend in these places.
If you just want a quick taste of 1 city in each of those countries so you can say you went, then yes, you can do something like Galway for 3 nights to spend time with your daughter who is over there, then Dublin, Edinburgh, London and Paris. You really don't have the time to devote to exploring other regions in some of those countries.