It has been a nightmare fixing things behind the scenes. They had put in some wrong IP addresses in one of the nameservers, so that might have been part of the problem with slow propagation. I went in there and fixed it myself. I've spent all day on this and still at it. The hosts support didn't acknowledge their error when I said I had to change IP addresses. Probably company policy to avoid any blame or liability.
Regarding the site speed, the hosting costs just over 1/3rd of the previous costs, but I was on an older server. I removed litespeed, which was supposed to be fast, but fast software running on a knackered old server leads to hardware bottlenecks. The new hosting has a faster processor and hard drives.