Is it possible to force Craft to recheck available sever modules? One of our servers hosting a Craft site has recently had ImageMagick installed on it, as we noticed that transformed animated GIFs weren't animating.

The 'Requirement Report' at /admin/utils/serverinfo is still reporting a 'warning' against ImageMagick (as it was before it was installed), and Craft seems to still be using GD as the freshly transformed animated GIFs still aren't animating.

Is there a way to force Craft to 'recheck' the available modules on the server (and so in my case, for it to recognise that it can now use ImageMagick)?
Craft will re-run the requirement checks simply by loading that admin/utils/serverinfo page.  The results aren't cached.

It sounds like your box may have had the ImageMagic library installed, but not the PHP wrapper for it.  You can check by going to admin/utils/phpinfo and searching for imagick and if there are no results, it's not installed.

Alternatively maybe it is installed, but you're explicitly setting the imageDriver config setting to 'gd'.  Be default, it's set to null which means Craft will automatically use Imagick if it sees its available.