From reading the rules on the website regarding geographical areas, I notice that West Yorkshire should be capitalised.

What's the rule if I wanted to write North-east Yorkshire, for example? I'm assuming the hyphen is correct, but what about 'east' - should that be upper or lower case?

You capitalise if it is an officially accepted geographical or other recognised designation of an area:

London's West End…

In the south-east of London.

East Midlands
West Midlands

The titles of current areas of civil administration are:
North Yorkshire
South Yorkshire
West Yorkshire
East Riding of Yorkshire.

So;
The dales of north-east Yorkshire

If…IF it were an official civil administration area, then:
North-East Yorkshire

Thanks Claire,
Thant's a nice, precise answer.