The upshot: in BrE one hires things (and sometimes places), employs people, and rents places; in AmE one hires people and rents things or places. That said, one hears hire for people in BrE too, but just not as much as one does in AmE. And employ is not particularly non-American, it's just overwhelmed by hire there. Both have let for what the landlord might do and lease for certain things (e.g. long-term non-ownership of cars, I think). It'll probably be easiest if I go through these verbs one at a time.
Rent can refer to the act of letting something to someone (I rented some land to him) or to the act of paying someone to use their something (I rented some land from him). This is old news--since the Middle Ages when it came into English from French. The OED notes one sense that is 'chiefly North American' which means 'To be hired out for or let at a certain rate', as in (their example):
1992 Albuquerque (New Mexico) Monthly Oct. 37/2 (caption) The tux, suitable for any performance in Albuquerque's doubtful performing arts center, rents for $55 and sells for $425.But why does AmE use rent for things besides (now particularly AmE) real estate and BrE doesn't so much? The first examples the OED has of non-real-estate rented things are American: a guide in 1817, boats in 1895 and pianos in 1903. Comparing rent a boat with hire a boat in American English via Google Ngrams, one can see how recent this change is:
So, use of rent for non-real-estate seems to be an American innovation, possibly motivated by more limited use of hire and/or by the advent of so-called rent-a-car companies in the 1920s.
I searched the Corpus of Contemporary American English (COCA) for nouns that came one or two words after rent a. The top 10 are: car, room, house, movie, boat, bike, video, place, canoe, kayak. Further down the list we get tuxedo, horse, harp and grandchild. Compare this to the British National Corpus, where the top 5 (because it is a smaller corpus) are: room, house, place, car, villa. (And half of the six rent a car examples are in the names of American companies.)
Americans can even rent time, for example (from the San Francisco Chronicle, via COCA):
He pays $10 an hour to rent studio time and pays to rent equipment when he goes on remoteWhile one can find British examples of rent studio time, the more common phrase would be to book studio time, using the much-more-BrE-than-AmE sense of book to mean 'reserve'. Book in this sense often gets extended beyond the action of reserving the room/time so as to include the using of the thing that was reserved.
A particularly British use of rent is noted by the OED (my emphasis):
But note that it's only rent-a-mob/crowd that is British. Rent-a-cop is label(l)ed as 'N. Amer. depreciative', and all of these humorous extensions have the American rent-a-car (BrE car hire) to thank for their existence.In various extended and humorous (typically derogatory) uses, suggesting the temporary acquisition or instant availability of the person or thing specified, usually for an expedient or mercenary purpose; spec. (chiefly Brit.) denoting a faction of regular, esp. violent, participants in public protests, in rent-a-crowd, rent-a-mob, etc. See also rent-a-cop n., rent-a-quote adj. and n.
Hiring people and hiring things both go back to at least the 13th century. So this is not a case of either nation making up new meanings, but of the 'thing' meaning dying out in AmE and gaining prevalence in BrE.
I searched the Corpus of Global Web-Based English (GloWBE) for nouns that occurred one or two words after hire a*. The * there allows it to be 'a' or 'an' (or 'any' or 'all' or anything else that starts with a-; other words are less likely to be frequent and therefore influence the outcome--but that doesn't mean it hasn't happened; see below). This is my way of looking for direct objects of hire. The software on the website calculates which words co-occur (or collocate, to use the jargon) with the search string in higher-than expected rates for each dialect. Here are the strongest collocates:
This is not to say that it's not real British English to 'hire a person', just that such uses don't stand out in the data. Hire in BrE is not a magnet for the word person like it is in AmE.
To give a broader sense of the kinds of things one can hire in BrE, the top 10 nouns after hire a in the (20ish-year-old) British National Corpus are: car, video, house, boat, bike, minibus, van, room, plane, helicopter. There's a distinction to be made here between hiring a room and renting a room. One hires a room for an event; one rents a room to live in.
In the case of fire in the BrE list, it seems to be that the verb fire has been mislabel(l)ed as a noun by the software that automatically tags words for part of speech. In this case, it represents the phrase hire and fire. So, that one is about doing something to people, but it seems to be part of a nearly-set phrase (it's used much more in the BrE part of the corpus than the AmE part).
Because hire is used so much, employ (orig. AmE) loses out in AmE. Searching GloBWE for employ a *man (which would capture employ a woman/a man/a postman, etc. but conveniently leaves out employ a metaphor or anything like that), I found 16 BrE examples and 0 AmE ones.
I've already covered this one briefly. Both AmE and BrE have this word with the meaning 'to rent out', but BrE has developed an intransitive sense that means 'to be let'. Thus one sees UK properties advertised as 'to let' where US ones would be 'for rent'. Click on the link to see what happens to 'to let' signs (if you can't imagine it).
To lease is the same in AmE & BrE. But I can't leave this post without mentioning that the British may get a new lease of life, while Americans get a new lease on life. Not a verb there, but if I hadn't mentioned it, someone would have asked for it in the comments, I'm sure.
Wrote this late at night, so glad to see a lot of good info on the fine points of employ/hire in the comments!