The process should be:
Put old SIM in old phone
Copy (NOT move) all SIM contacts to old phone
Put new SIM in old phone
Copy (NOT move) all Phone contacts to new SIM.
Put new SIM in new phone
Copy (NOT move) contacts to new phone which should then sync with a Google account so that you never have to do this again
(always copy, never move, in case something goes wrong).
So let's understand why you cannot put the bits together to make the nano SIM the size of the old SIM. They should just press together. IF you've broken them, then you can buy them again.
This is just one example, you can get similar in any phone shop and certainly Vodafone used to give them away free.
www.amazon.co.uk/Ciscle-Adapter-Converter-Micro-Standard-Black/dp/B01D4DNE8O/ref=sr_1_4
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