Welcome to ja.com
My name is ***** ***** i hope i can be of help to you today but first i just need to ask you a couple of questions so my answer is better sourced for you.
1. When you say the alternator belt is turning, is this just while you are turning it by hand or when you are trying to start the car also?
2. If it is also when you are trying to start the car, is it a fairly quick kind of whirring noise?
I look forward to your replies
Thank you in advance
If it has started using the second key at least that means that the starter is ok and also means that nothing worse has happened like the timing belt snapping etc.
What you should do though is check the battery in the first key as this may need replacing and this could be why it is not being recognised by the NATS system. Once it is replaced you will probably find that it works again.
The NATS systems are plagued with problems and always have been. My own old Nissan used to do a similar thing with starting as it was not recognising the key and then the central locking started going haywire.
I hope this little bit of extra info helps you some more.
If you require help in the future please don't hesitate to contact me on this site via my profile and i will be happy to help you.