Local SEO without a local Address visible
You can hide the address on Google and it can still rank but you are
still subject to Google's local ranking algorithm, which a major
factor is trust and this is linked to having your business address
mentioned on various 3rd party websites such as Yelp, Yell, BT,
Hiding the address completely on Google, Website and 3rd partys will
have a negative impact and will make it much harder to increase those
rankings. This is because serious businesses regardless if they are
small, big or one-man employed businesses they should have a visible
address for customers to write complaints to, or report for fraudulent
activities. After-all, as a consumer do you really want to hand over
your hard earned cash to someone that has no address...
Google, and customers don't agree with the 'I don't want my address
online' excuse, because if your a business you should have a mailing
address at least. Some sole traders like plumbers will use their
accountants address for mail.
Some people and businesses will add citations to various sources with
Subject to appointment only statements. I use this for my own business
on HotFrog for example, I seriously recommend that you list your
business address on both the website and various reputable sites
Businesses that have physical registered and visible addresses are more trusted by customers and this echoed into the virtual world. Both Google and Bing rewards site for visible (NAP).
If you want search engines rewarding you fully, in the region that you are targetting then you must have a local, visible physical address in that region, or at least visible address on 3rd party websites for NAP.
Also, if your address is registered on the out skirts of London, even if its not visible to customers and you select a radius of 100 miles, then the likelihood is that Google will still favour customers nearer to companies nearer to them, even when you have the address hidden! remember, your address is hidden to Customers, not Google, proximity will still apply, increase your local rankings, increases the proximity rating.