Changing a SIM card shouldn't affect your ability to log into your account because passwords and/or account details aren't stored on SIM cards. They're only stored within the phone's memory. So I'm largely mystified here.
As I read your post, you can still access the relevant mail account, so you should be able to enter your email address on the log-in page and then click on 'Forgot Password'. That will result in a new password (or instructions for creating one) being sent to your email address.
Otherwise try an email to
[email protected] or a phone call to 0333 444 1280. (Calls are charged at the same rate as those to 01 or 09o2 numbers, from both landlines and mobiles. That means that, if you're using a 'bundle' with 'included minutes', it will be free)