Not quite as tough as putting a 6.7l Cummins diesel in a volkswagen BUT....

So much has been done since 1.5.5f (including some security updates) that it is actually safer to upgrade.

Put 1.5.7d in a new/test folder. Bring over images, define pages, and your template IF compatible. Put current versions of mods in the new site and import/update the old database.

All can be done while the old site is running.

Meanwhile, being in the UK, you do not mention which PayPal you are using and whether or not your host has changed PHP versions on you.

Do a google search for PayPal UK Zen Cart and look at some of the current threads on what's going on in your area.