Q. Is it all right to ask guests to remove their shoes? I hate cleaning up the dirt, salt, and mud that get tracked in.
For a casual get-together, it's fine to ask guests (nicely) to take off their shoes.
P.S. If you can, try to have extra pairs of nonskid socks or slippers on hand for those who feel uncomfortable (or embarrassed) exposing their bare feet. Say, "Would you like to trade your wet boots for some warm, dry slippers?" But for a more formal event, like a dinner party, where guests are dressing up, it's most polite to let them keep on whatever footwear they want to wear.
This is basically the moderate shoes-off position- shoes off, except at parties or formal occasions. Personally, I think shoes off at a formal dinner party is fine. Millionaires and celebrities wear tuxedos and ball gowns at yacht parties and still take their shoes off.
I must also express my doubts about the wisdom of offering slippers in the UK. Most British people would just think that is too weird and would rather be in socks or bare feet.