Realistic? It's possible to do it. Enjoyable? Smart? Doesn't look like either to me, not with 13 days (and you should really detail exactly how many days you really have, not including arrival and departure days).
Remember, every time you pick up and move, that burns most (or all) of a day. So when you say X days here, Y days there, you need to subtract 1 from all those numbers. So you get 2 days in London, 2 days in Dublin - oops, no, one day in Dublin, less than one day in belfast, 3 days "Galway area", one day in Glasgow - oops, no, less than one day each in Glasgow, less than a full day in Edinburgh...
That looks like a punishing death-march to me.
Start by being honest with yourself: how many days do you have, not including arrival and departure days (no matter what time flights are scheduled)? However many days you have, you have three countries (four, actually, since Northern Ireland should count as one). I think you need to cut out one of them. Knock out either Scotland, or Ireland. Sad to say, Ireland does not look like a good fit here. Take (all of Ireland, the Republic and Northern Ireland) that out of the equation, and it starts to look more realistic. With 13 days, you can get a quick, but enjoyable, taste of some of the UK.
I'd start in London and work my way up to Edinburgh.