Yes. We bought a smartphone in Tesco, found the SIM from the old one was the wrong size, so the assistant took it away for a few minutes and transferred the phone number from the old onto the new SIM. The new SIMs are multi-size ie, they come as large size and you snap-out the middle to get a small size.
Please note that you cut your SIM card down at your own risk.
We've cut down SIM cards a few times, and it's worked in all but two instances. Those aren't actually great odds. And one instance was a bit of a headache because the SIM card could no longer be used at all (not even in the original phone).
Please remember that there are always risks involved with home modifications and don't cut up your SIM card unless you're happy for it not to work (in this case you'll need to get a replacement from your carrier or buy a new one). Proceed at your own risk."
The old phone was PAYG, the new one is a fixed monthly payment covering an amount of internet use and a number of calls/SMS. You can specify a sum by which you can exceed your monthly allowance if you wish. Both old and new phones are tied to the Tesco network but I don't think that had any bearing on the fact that the number could be transferred from SIM to SIM. The credit on the old phone at exchange-time was transferred to the new one, so the first few monthly payments were adjusted accordingly.