I'm having a similar problem, although for me this is a 'first time of trying' situation.
The site I'm working on is set up as follows:
Shared hosting.
Shared SSL
Prestashop installed in:
domainname.co.uk/shopfront
(hosting package required a folder to be specified for SSL (slightly annoying)).
In Preferences>SEO/URLs, the default details are:
Shop domain - domainname.co.uk
SSL domain - domainname.co.uk
Base URI - /shopfront/
If I click on "Please click here to use HTTPS protocol before enabling SSL", I get the same basic problem that Adam descibes above; the https appears in the address bar on that page but the option to "Enable SSL" doesn't appear. The admin remains in https but the site stays 'normal'. If I manually add the "S" the site simply refreshes to 'normal' status.
Just as a sanity test, I threw some things in the basket and went to Checkout, but no change to https is happening. I need to get this sorted out before spending any time on styling and filling the shop with goods, so any help would be most appreciated.
Adam, are you able to expand upon the solution your hosting company provided? I'd just like to be armed with a bit more information if it comes to me having to call the host in this case.
Are there other reasons why this might happen?
Many thanks for all assistance offered.
Frank