Because the differences between British and American English usage of rent and hire wasn't really dealt with, I did some research and came up with this rather detailed summary.
- Rent (the fixed amount of money you pay to an owner for the use of something, especially that you pay regularly for; a room, a home, an office etc.)
As noted by RyeBread, Zibbobz and tehDorf, in the USA only goods and accommodation are rented; in extremely rare cases are they said to be hired. The contract to rent an item or property can be either short or long term, for example: rent a car; rent a Halloween costume; rent an apartment; rent a movie; etc.
Rent accommodation (out) to
The owner of the property rents out to tenants, the sign displayed outside a home advertising its availability will have the words: House for rent.
The consumer (or lessee) also has the opportunity to purchase the rented good(s) by paying the rental fee for a length of time or by paying a lump sum payment. This form of purchasing is known as rent-to-own or installment plan.
In the UK it is more common to rent goods (TVs, furniture etc.) and accommodation on a long term contract.
Rent accommodation out / Let
Houses, flats (apartments AmEng) etc. which are rented are usually let in the UK. A home displaying a sign in the window might have the words: Rooms to let. A British home owner might say: "We live in the downstairs flat and let the upstairs one to tenants." Let and rent are nevertheless, both common terms used in the UK.
an agreement where a person pays for an object or property to use for a short period of time.
As previously mentioned in the question; goods in the UK can be hired or bought on HP (Hire and Purchase). Regardless of its size, any object can be hired in the UK be it a bicycle, a DVD, a room to host a conference or even a castle. But the arrangement is usually a temporary one. E.g; Windsurfing and water skiing equipment on hire; "prices include return flights and car hire"; "they hired a marquee for the wedding". And clothes are hired, usually for single occasions, not rented.
- Hire (to employ or give somebody a job)
In both the UK and in the US the expressions hire and employ are used for employing people on part-time or on permanent contracts. The small difference being that the expressions take on, and employ are used more frequently in the UK compared to the US.
@Barrie England's answer (see link) offers a further alternative: "In the UK, we might be more inclined to appoint a consultant."
Main source: Cambridge Business English Dictionary