Unfortunately, I don't think it's as simple as "just use the 'url' and 'email' input types for their respective fields."
Using the "url" input type can be a problem, due to strict validation - the validator requires a URI as specified by RFC 3986, meaning that if the user enters "example.com" rather than "http://example.com", the browser considers this invalid, blocks the form submission, and displays an error like "Please enter a URL".
This effectively breaks the form for most users, on mobile or not, so you'd need to disable HTML5 input validation on this form or field.
Good point, however, I'm not sure I can recall a time during which someone intended to collect a URL from me but allowed just "example.com" as a valid input. I'd venture to guess that the validation enforced more often than not actually helps the site owner achieve their intended goal.
Using the "url" input type can be a problem, due to strict validation - the validator requires a URI as specified by RFC 3986, meaning that if the user enters "example.com" rather than "http://example.com", the browser considers this invalid, blocks the form submission, and displays an error like "Please enter a URL".
This effectively breaks the form for most users, on mobile or not, so you'd need to disable HTML5 input validation on this form or field.