SOLUTION!
Everything changed when I read this article:
Why can't I send email from wireless hotspots using a laptop computer and Windows Vista's Windows Mail?
I can't post a link so you're going to have to Google.
I had always been given to believe that you had to use the outgoing mail server of your ISP not of your e-mail provider. Furthermore, it worked with Three (smtp-mbb.three.co.uk) until last week, but not at all with Virginmedia, my home broadband supplier. They had once told me that they don't support 'relaying' at all, with the result that even at home I had to use my mobile broadband connection to send POP3 e-mail.
After reading the above article, I tried resetting my outgoing mail server (on laptop and iPhone, across two totally separate e-mail accounts) to the mail account provider's outgoing mail server settings and selected password authentication using my incoming mail server user name and passwords. BINGO - I can send outgoing mails on whatever connection I like - even the Virginmedia home connection.
Why was this so hard to discover. Over the years I've had numerous calls with various support people on this issue. Even this week, the guy at Three (who seemed pretty competent) never said "why don't you use your e-mail providers outgoing server with authentication?" INCREDIBLE.
Hope this helps other non webmail users.