The IP details that Google and the like use is far more detailed than just where your ISP is based. If you look at Google analytics for your site you will be able to drill down to city and in some case IP can even go down to street level. Do you ever see ads for online dating that say “meet people in Bristol”. This is using the IP of your computer to know where you are.
The best thing is to have Chester in your keywords, Chester in the meta data of your webpages and Chester in H1 and H2 tags of your webpages. If Google ads are then setup for a specific location then you should be alright. Testing adwords is always slightly hard as then never show to everyone unless you have some massive campaign that is buying all the ad save or you are very very niche.
What you could do to prove things are working is let the ads run for a week or so and then use Google analytics to see where people are looking at your website from. If they are not motley clustered around Chester then I guess you have a problem.
It is all a bit of minefield that I am not 100% sure about, not sure anyone is.