After registering with PayPal, you should have received a confirmation by email from PayPal that your payment was completed.
Usually you should then receive your registration information very quickly, but it can sometimes take several hours.
If you still haven't received your registration key, please check that your PayPal mail address is correct (sometimes it is incorrect, obsolete or refer to a full mailbox). Especially after you created a PayPal account, the email address you used may no longer be valid but still be used.
Check also that the registration mail was not considered as spam and not moved to the "Junk" folder. Check Junk folder in Thunderbird but also the Spam folder of your email provider on the web. For instance, check the Spam folder of your Gmail account on the web. Also, sometimes the registration mail doesn't reach the recipient or are returned with an error. Maybe because of too zealous mail servers which consider it as spam... Spam is a plague not only for people who receive mails but also for people who send mails.
So if you still haven't received anything, please contact sales [at] birdiesync.com with PayPal transaction id, provide an alternative mail address and mention that you check that the registration mail was not moved to the spam folder.