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.