We haven't run out... there are millions and millions of spare numbers - just not within the current area codes. In the case of Bournemouth, usable numbers beginning 01202 are running out.
There are plenty of spares elsewhere: you could give Bournemouth a batch of numbers starting 01660, or you could renumber the place to use seven digit numbers and a shorter area code e.g. (0119) xxx xxxx. However, Ofcom believes that people want to stick with their familiar area codes, rather than see several different area codes covering the same area or getting caught up in renumbering.
The USA and Canada have been using a different strategy - they just issue extra area codes for the same area when numbers run out. So, New York originally had the area code 212 and 7-digit local numbers. Over the years it's evolved to the point where a New York number could start with any of six different codes: 212, 718, 917, 347, 646 and 929.