Gateway Timeout error when clicking to payments

Hi,

The error message indicates that there is some kind of proxy between you and the server and that while the current page is being processed, the proxy reaches its timeout limit, and returns you that error message before your server sends its response to it.
So that doesn't say what is the problem on the server (since you don't get back the error from the server).

However, at the end of the checkout, the only things that can run for a long time like that for any payment method selected are:
- a third party extension or a mass action looping endlessly.
- the connection to the mail server hanging when sending the notification emails

And if it happened just after the migration to the new server, it's probable that the email settings of either the Joomla configuration or the HikaShop configuration have to be adapted to match the requirements of your new hosting. Something is probably off there resulting in that hanging.
To test that, disable all the emails in the menu System>Emails and try again. If it works, that would confirm my theory and you'll have to check your email settings with your hosting. If it doesn't, then it comes from something else and I would recommend checking the error logs of the server (apache, PHP, etc) in order to get some clues as to why this is happening.