If I have been using a certain brand name for my products and services for some time, but haven't registered it as a trade mark, would someone else that copies and registers it be able to prevent my use of it and would I be able to stop them from using it?
This is a trade mark / passing off question, rather than a copyright question.
Can the owner of a registered trade mark prevent a business with prior unregistered rights (such as rights in the tort of passing off) from using that mark?
Yes, in some circumstances.
There is a special defence in Section 11(3) of the Trade Marks Act 1994 which may protect the business with prior rights, but the defence has its limits:
A registered trade mark is not infringed by the use in the course of trade in a particular locality of an earlier right which applies only in that locality. For this purpose an “earlier right” means an unregistered trade mark or other sign continuously used in relation to goods or services by a person or a predecessor in title of his from a date prior to whichever is the earlier of - (a) the use of the first-mentioned trade mark in relation to those goods or services by the proprietor or a predecessor in title of his, or (b) the registration of the first-mentioned trade mark in respect of those goods or services in the name of the proprietor or a predecessor in title of his; and an earlier right shall be regarded as applying in a locality if, or to the extent that, its use in that locality is protected by virtue of any rule of law (in particular, the law of passing off).
You might be able to use your prior rights (via passing off) to prevent someone from using the mark. Also, you might be able to block a registration under Section 5(4) of the 1994 Act:
A trade mark shall not be registered if, or to the extent that, its use in the United Kingdom is liable to be prevented - (a) by virtue of any rule of law (in particular, the law of passing off) protecting an unregistered trade mark or other sign used in the course of trade, or (b) by virtue of an earlier right other than those referred to in subsections (1) to (3) or paragraph (a) above, in particular by virtue of the law of copyright, design right or registered designs. A person thus entitled to prevent the use of a trade mark is referred to in this Act as the proprietor of an “earlier right” in relation to the trade mark.
However, stopping use after the other person has a registration could be problematic because of the effects of Section 11(1):
A registered trade mark is not infringed by the use of another registered trade mark in relation to goods or services for which the latter is registered (but see section 47(6) (effect of declaration of invalidity of registration)).
UK registered trade marks are not very expensive for most businesses, and if you are concerned about these issues I suggest that you consider applying for registered protection asap.
(The answer above assumes English law would apply. Different rules will apply in different jurisdictions.)