In the UK there is no registration of copyright; it exists automatically by virtue of something having been created that attracts copyright (see our factsheet
here).
To prove copyright, evidence of ownership and creation is required, that can be done by keeping signed and dated copies of the material produced at the time.
Generally speaking, the author of the work (or his company if he is an employee) will be the first owner of copyright in the work unless there is something expressly to the contrary.
A notable exception is where a logo is designed for somebody's exclusive use to identify their business (see the
Doc Martin's case), as it may then be ruled that an implicit term of the contract exists for copyright to belong to the client company and not the designer - on the basis that the transaction does not make sense otherwise; nobody would assume that it was intended that the designer should retain copyright.
It is sensible, however, not to rely on this (as going to court to prove a point can be expensive), but to make sure that when a designer is instructed it is agreed in writing who will own the copyright - any reputable designer will agree to this and that copyright will pass to the client when the designer's fees are paid.
Copyright only protects against copying and not accidental similarity - if I create a similar looking logo and did not know of your logo, that is not copyright infringement. A commercial logo may therefore also be registered as a trade mark, which will protect against any confusing similarity even if there was no copying and give you a certificate and public proof of ownership on the trade marks register (see our factsheet
Protecting Your Identity).
However, what is always more important to protect at the outset by trade mark registration is the name of your business, as registration of a name will protect against confusing similarity in any form - even if the logos are different - and it is much more likely that someone might come up with the same or similar name than that they might create a similar logo by accident.
Trade mark registration for your name is also a powerful safeguard as protection is granted from the date you apply - even before you use the name - and gives you protection across the UK and can stop somebody using the same name for the same business even if there is no confusion.
However, the name must be distinctive, i.e. recognisable as a brand and trade mark; you cannot for example register LONDON GARAGE to cover garage services in London (although a logo might be registrable).
For anyone needing a logo designer, I can certainly recommend
Eagle from personal experience; other reputable designers exist of course too.