Getting A Business Website To The Top?

I created a website for a business and it is slowly working its way up google but I'm just wondering about how long you would estimate before it makes its way to the top and replaces the old website?

The website is for a local appliance repair business. Currently the website that is at the top of google with related searches is a site by Yellowbooks. That site will soon be down and I'd like to know how long it will take before the new one takes its place. It is currently on the 4th page of the google search "appliance repair (name of local area)".

 

Does the old site include a link to the new site? Also, will the old site eventually redirect to the new? The key to maintaining the Google position lies with the number of visitors so it makes sense to have these redirects in place.