What do you mean by it stopped working? The button doesn't submit the form at all? The page just refreshes? Please be more specific. Going from J2.x to J3.x means your Joomla template is no longer compatible and probably is a template issue. Try switching to default Joomla protostar template and see if issue persists.

If the button does nothing then jQuery is erroring. Meaning you've a fatal JavaScript error on your site. Press F12 to open your browsers developer tools, click the Console tab, and check for any errors (they'll be red).

Did you turn CBs jQuery off in CB > Configuration > Integrations? If so turn it back on. If it's already on and you know for a fact your site provides jQuery on every page you can optionally turn it off there as it sometimes conflicts with template jQuery loading or 3rd party jQuery loading extensions.

Template is most probably not the problem, since I have other test site locally with the same template (but basic data, not the whole site) and the reg form works there.

I've heard this probably a 1000 times over the years and you could be right, but please just try switching to protostar template and refreshing the registration form then retrying.

The new is that I am now getting the error message while submitting form - "The email address you entered is invalid. Please enter another email address.". I have tried different mails, none are accepted.
I've checked the validation rules for that field,- all looks normal...

Probably the MX Record lookup check failing. Within CB > Field Management edit the Email field and under Parameters > Validation set "Enable Email checker" to "No" then see if registration works. This is supposed to be a non-blocking check though and will look into why it's failing.