It takes somewhere between 6-10 weeks for new website addresses to be indexed by search engines. Once your new address is picked up and catalogued it will steadily rise through the rankings so patients will be able to find it easily. There is nothing you or My Surgery Website can do to speed this up, just a little patience is required.
If your website has been live for more than 3 months and is still not appearing there are certain things you can do to help. You, of course, cannot top the rankings for all searches but the key element in all searches now is relevant content. If the pages on your website contain the text you think your patients will search for this will really help with your rankings on search engines. Also it is of great importance that your site is updated regularly. The search engines put a lot of emphasis on relevance and a site that is current and up to date will rank higher than one that is not.
We definitely recommend reading this Article: The DEFINITIVE guide to SEO on your Surgery Website