First of all you do not need " coinage" for tipping in dollars.
A minmum tip would be one WHOLE American dollar. No one would tip less as that would be an insult, even if all room service brought you was tea, tip would be 1$ min. you should tip 2 or 3 dollars min if they bring a full meal.
I am sure British Banks have American dollars.They seemed to last time I was in England 4 yrs ago, perhaps they do not now, but I doubt that, are you perhaps from a small town? A larger city may have what you need, remember thousands of American tourist dump their dollars in your country, so they gotta be floating around somewhere!
Since you do not need many perhaps try an exchange booth at the airport( I know the rate is not good but you don't need that many to start with, just get 10 or 20 pounds changed for handing out in dollars.
I am not familar with Euros, but I imagine since alot of your tours/ports are going to be to countries where the currency is Euros, then you should tip in Euros. Just get a bit to start off with, and really how many tips do you really need?
We are Canadian and don't quite get the tip tip tip thing, I mean we do tip room service, and we leave the shipboard tips on our account, but we rarely tip anyone else. Perhaps a tour leader, only if he was very good though, those shore exes are so exspensive that I do not feel the need to pay extra unless it was an especially good tour. and of course we tip a taxi driver, but it is usaully just added to the fair( rounded up) .
I realize that your coins come in bigger denominations , but the biggest in American ( that is widely available and used) is a 50 cent piece. Most common coinage is 25 , 10 and 5 cent coins. Not great tipping money. I recommend you not get any coins at all.
Anyways have fun and don't let the whole tipping thing drive you nuts.