If she is with O2 then she could have been moved on to the simplicity 30 day contract, this is basically a contract (30 day one) where if you are happy with your phone you can use the Simplicity tariff and by doing this it costs less as there is no phone supplied with the Simplicity tariff.

You should be able to use her "cast off" on pay and go.

Regarding "whether on the same network, or another. Would this be possible or would her provider object??"

Basically once you have completed the contract the phone is yours to do whatever you want with. Some contract phones are basically able to accept any sim card from any network, as the user would have a choice to go on to either Vodafone, O2, Orange, so the phones have to be made to work on the various networks. However, if she got the phone from say for example Vodafone shop, Orange shop, the phones will only work on that network. You can however in most instances get the provider lock removed.

If she has purchased the phone from say for example Carphone Warehouse then the phone should accept any sim card, unless it is a model that is exclusive to say the Vodafone network. I believe that the O2 shop supply phones that are not locked to their network, although I am not 100 percent sure.

Best thing is to try your sim card in your cousins old phone and see if it works.