Hi S. -
DO NOT bring an expensive stroller on your trip. You can't bring it on the plane with you (you'll have to gate check it) and the airlines are not held liable if there is any damage to it when you arrive. I know from experience. I had always traveled with my Maclaren stroller (which I love!) and on my last domestic flight with my son, it came up from gate check with the sunshade broken on one side. When I brought it to the attention of the gate agent, she said there was nothing they could do since it's not covered under the luggage insurance policies. I still have my broken Maclaren, but that $200 stroller is now my travel stroller and we bought another Maclaren for everyday use around town.
I've found when traveling with my son (and he's already earned a free ticket on several airlines with his frequent flier miles) that the less stuff you have, the better. We do take his car seat with us because he is much more comfortable on the plane in his own car seat, but we usually just leave it at the hotel when we arrive and don't touch it until we are leaving again (or install it in our rental car). You can buy a travel bag to carry it in off of Amazon. The one I have was $15. Don't bother trying to install a car seat in a taxi. It's not worth the time, and taxi drivers don't like waiting (unless they are on the clock!). All of the taxis I've taken with my son - even in NYC - the drivers recognize there is a small, unrestrained child in the back seat and they have driven more carefully. I've only ridden in a taxi a handful of times with my son, and he's always sat on my lap. No problems.
As far as the pack 'n play, can you contact the hotels where you are staying to see if they have cribs available for you to use? Pack 'n plays are convenient, but they are terribly heavy and a pain to travel with. Most hotels have cribs for their guests, you just have to let them know ahead of time and they'll have it in your room when you arrive. I always bring a crib sheet from home rather than use the hotel sheet. I'm not sure if it does anything, but it gives me peace of mind and it gives my son a familiar feel from home.
Remember to bring a change of clothes for both you and your daughter on the plane. You never know when spills may happen on either one of you. I bring a new toy or two for my son to discover when we fly, which is always a big hit. Crazy as this sounds, some of the best toys are easy toys... like dental floss (pulling the floss out of the package) and Band-aids (like stickers, but easily removable and cheap). Don't pack as much clothing as you think you'll need. You can always do laundry at the hotel and pick up things while you are vacationing. And bring as many diapers as you'll need for the plane ride and one day overseas, and then just buy diapers when you arrive. It may be more expensive, sure, but it's not worth carrying loads of diapers in your luggage.
Have a great time in the UK and Ireland! Enjoy your vacation - it sounds lovely!