Greetings
I had exactly the same problem recently.
Two choices, 1, don't worry. The numbers will not disappear of the old sim card so as long as you keep the old card in the old phone you will be able to transfer them manually at your leisure.
2, Go to the local mobile phone repair shop, for a small fee they will effect a speedy transfer or sell you a sim card reader (or eBay but don't forget it will need to take both sim sizes).
Lastly, go through the pain barrier, use your address BOOK it is the ultimate backup.
Good luck, cheers, Ian
Ian, August 2016