Not having a registration number will not keep you from being able to use IRC-- you should be able to freely connect to any server you wish even while not registered, just close the nag screen.
As far as your registration #, rest assured, it is likely on its way-- email is theoretically instantaneous, but not always.. check your junk box to see if it maybe got accidentally flagged as spam, and if not, wait-- in rare cases (keyword: rare, this probably won't happen) it can take a day or so, just use mIRC unregistered until then.
Your mIRC registration information was emailed to you a few minutes after you registered - if you did not receive your registration email, it is probable that your email program treated it as spam and deleted it. Please see the registration section here to find out how to retrieve your lost registration.