Tricia,
These questions are more about how Google and other search engines work than about who provides your hosting. They look around all websites (with some exceptions) periodically and record what they found the most recent time they looked.
If your former host still has material relating to your domain or business, a search engine will find it. If you can log in to your old server, delete all the pages or leave just a link to your domain home page so that anyone finding the old site will eventually find your current one. If you cannot still log in to your former host, only their staff are able to delete the old stuff from their server. You would need to ask their support staff to do it for you. It may not be instant but they usually do it if asked nicely.
As for the half-built site, does that mean you put some of it live before it was all finished? It may be worth checking your BT server to see whether any pages with names you no longer use are still there. Delivering a new site does not delete old pages if they have different names.
Once you only have up-to-date material visible on your webservers, the search engines will soon catch up and direct people only to that material. Allow a few weeks. It is salutary how long anything that has ever been put on the internet may remain publicly visible. If you want to test things without having the whole world find them, try looking up how to use files called robots.txt and sitemap.xml. They don't make things completely private, but they do instruct search engines on what to look at.
I hope some of this points you in the right direction.
Nigel.
