Two Farthings = One Ha’penny. Two ha’pennies = One Penny. Three Pennies = A Thrupenny Bit. Two Thrupences = A Sixpence. Two Sixpences = One Shilling, or Bob. Two Bob = A Florin. One Florin and One Sixpence = Half a Crown. Four Half Crowns = Ten Bob Note. Two Ten Bob Notes = One Pound (or 240 pennies). One Pound and One Shilling = One Guinea.
The British resisted decimalised currency for a long time because they thought it was too complicated.”
-- Good Omens, Pratchett & Gaiman
Since the divisibility argument is often trotted in defence of continued use of non-metric measurement systems, it seems odd it's more rarely used to defend non-decimal currencies, where it might actually be useful. Though the fact nobody uses meaningfully-non-decimal currencies any more is some evidence for arguing that this just isn't as useful as the convenience of the easy decimal numbers.
Still, however many currency sub-units you have, inflation makes the issue less pressing over time.
Now it's pounds and pence which work very much like dollars and cents. 100 pennies to the pound.
The biggest difference now is probably that Americans have cents, nickels, dimes, and quarters, where the British have pennies, two-pennies / two p (sometimes, but rarely these days, called tuppence), five p, ten p, twenty p, and fifty p.
To me that's easier to keep track of than remembering which one's the nickel and which is the dime.
The only term still in general use seems to be 'quid'.