On most server configurations the same files are used to serve https or http. Ideally when you add https certificate you also add a .htaccess rule that redirects http requests to https. If you have a Cpanel hosting control panel they actually provide free certs but your host can enable/disable it. Minimally you should have option to install the cert yourself which is not that difficult. You can obtain one for free from service like Lets Encrypt.
If your host does not at least provide the option to install your own cert I'd suggest finding new host.
The biggest issue about waiting is links that are posted in your forum and elsewhere. Those can be handled with the redirect. Optionally the links inside of posts can fixed with database query.
When you switch it you need to go into phpBB's control paenl and underr cookie settings enable secure cookie. For server server settings use:
Server protocol: https://
Server port: 443
“Results! Why, man, I have gotten a lot of results! I have found several thousand things that won’t work.”
Attributed - Thomas Edison