Moving WordPress websites from one hosting company to another is a service we provide, while not normally a difficult task it does require a little bit of thought and planning, the typical cost is £60 ex vat.
The starting point is to make sure that WordPress and all its plugins have been updated and the site is working normally, then download the wp-content folder and take a copy of the wp-config file, finally access the hosting control panel and take a copy of the database (MySQL Dump).
On the new hosting server create a blank MySQL database and import the data from the old hosting. Install a fresh copy of WordPress to avoid any issues of hacking or malware that maybe present on the old hosting.
Upload the wp-content folder and edit the wp-config file to access the new database. To test the new hosting edit the Hosts file on your PC to browse the new hosting using the site’s domain name. Assuming that the site and any email forms are operating normally the domain name changes can be made to switch over to the new hosting.