Hi Laura,
I used to know someone who was also married to an EU citizen and living in the UK and she did have to get a spousal visa to work here once she arrived. So you will probably need to go to Croydon, I'm afraid. :-/
The Home Office is definitely who you will need to check with tho.
By the way, just to be clear, a National Health Service (NHS) number is not the same as a National Insurance Number (NIN). The NHS number comes to you by registering with your GP and is only used for NHS stuff like hospital visits, etc. The NIN number is more like a social security card...it has to do with your tax rate, unemployment, etc.