How can I find out what the art market is like in the UK for buying
contemporary art? (private individuals and companies).

Hello Clarette, (I sounded a bit like Hannibal Lecter then..lol)

There are many great sources for finding out about the UK contemporary
art market but the best source must be the Contemporary Art Society.
Since on their web site they confirm.

"We promote the collecting of contemporary art through our gifts to
public museums and the advice and guidance we offer companies and
individuals"

They are a charity supported by the arts council and one of their
mandates is to help people like yourself, so I am sure if you email
them, they will be happy to provide further assistance.

Contemporary Art Society
17 Bloomsbury Square
London WC1A 2NG

0044 (0)20-7831 7311
Email: cas@contempart.org.uk

Contempary Art Society Web Site. 
www.contempart.org.uk 

Other Sources
*************

There are a whole host of various other sources for your needs on this
page that I found at

The Guardian
http://www.guardian.co.uk/consumer/story/0,3605,641434,00.html

There you will find lists of up to the date magazines which give news,
sales information and advice from a variety of sources such as

Art Monthly
www.artmonthly.co.uk

Art Review
www.art-review.co.uk 

The Art Newspaper
www.theartnewspaper.com 

There are directories listed there for information and contact
information for various galleries.

And the most relevant one for your question is the listing of trade
associations and their contact details and web addresses.

Sources
*******
The Guardian
http://www.guardian.co.uk/consumer/story/0,3605,641434,00.html


Search Keys
***********
"Contemporary Art" and Markets UK search "Art Market" Search within
results = contemporary" "Buying Contemporary Art" UK



I hope this helps

Johnny Phoenix

Good answer with very useful links to other websites.