I travel solo so some of my recommendations might need revision depending on the number of additional travelers (significant other/children). I've done approximately a dozen trips to the UK/Ireland since 2010, including a six-week journey around Scotland in 2014. I have not needed more than 50 pounds in currency on a trip to the UK or 50 euros in currency on my trips to Ireland, and I always have currency to spare at the end of my trip. I hang on to any leftover cash since I know I'm returning to these countries fairly regularly but if you want to convert your money, don't use Travelex or some similar service at the airport where you'll probably pay a conversion fee. You'll almost certainly find a Starbucks nearby just before you leave, and many have cards unique to their country. Get one last souvenir, put your remaining money on your new Starbucks card (it'll convert itself to US currency), and wait for the question from your barista back in the US asking about where you got your cool card. Your local bank or credit union or AAA branch will provide pounds or euros before you leave if you want the security of having the currency in your hands before you leave, but if you do that, I really wouldn't take any more than 50 pounds or euros, and I suspect that 25 would tide you over until you get to a city ATM.

Credit card acceptance is much more universal now than when I began traveling sporadically to those countries 20 years ago. Bank ATMs are generally available, even in small towns of 1000 or so. If your travels will focus on the bigger cities, which appears likely given two days in Ireland--suggesting Dublin--and six in the UK--suggesting London for at least part of the trip--you should have no more problems finding an ATM than you would in the US. If you travel to a small town or are part of a tour group doing an excursion where you leave the driving to someone else, it may help if you get money before you go there, just in case a town's sole ATM is not working or the bus driver doesn't take you anywhere near an ATM.

My biggest caveat in regards to credit card use in European countries is to keep all spending in the local currency. You'll find that stores, restaurants, etc. will ask if you want the bill totaled up in dollars or the local currency. Accepting your bill in dollars will include a 3-5% fee tacked on for the convenience--which they will not mention. Since we now can monitor our credit card spending online fairly easily, including seeing the dollar amount spent on a particular purchase, I see no need to tack on an extra fee so that you know immediately how much you just spent in dollars. I actually create a fairly liberal daily budget in my head for expenditures in the local currency (current exchange rate for one pound is 1.3, for one euro, 1.16) and just carry over the unused amount to the next day. That keeps me from accepting the dynamic currency conversion offered because I'm already thinking in terms of the local currency. By the way, although I haven't seen this in my travels, I recently read an article talking about some bank ATMs starting to offer a similar currency conversion when you withdraw cash--with exactly the same additional fee for the conversion. Obviously, you don't want to pay that either. Of course, you should be sure the credit card or cards you use over there do not charge foreign transaction fees, which may range from 1-3%.

As gfunkdave mentioned, Visa and MC are more generally accepted than Amex and Discover although I am finding Amex more accepted than 20 years ago. Credit cards are also accepted pretty generally in small towns, but there can be instances where cash is the only form of currency accepted. I found this as recently as 2014 when the lone service station in a small town about 30 miles north of Aberdeen took cash only for gasoline.

I've found that tips of 5-10 pounds per person for a tour guide are gratefully accepted, but I'm no expert on this. Remember that in most restaurants, tips are not required as the servers and bartenders are paid a much more generous wage than in the US. In fact, in 2012, I tried to leave a bartender at a pub in Dublin a tip, and he looked at the money on the bar and said, "what is this?", pushing the money back towards me. Even if a restaurant includes a tip line on a bill (and many more of them do nowadays, probably in anticipation of American tourists), I've read through several sources that 10% is fine.

Hope this proves helpful. Have a great trip!