We've looked into it, and we're engaged (not for that reason, and being engaged makes no difference)
- but in Denmark there's no real difference between married and unmarried couples.
There's also a Danish rule and an EU rule for things in most cases, and you can choose which way you want to do it if both are open to you.
Under the Danish rule, we would have to have lived together for two years and be able to prove it (as far as I can tell), and we've only lived together since September. I can't see any way for us to meet this requirement if they ask me to leave at the 6-month point.
Under EU rules, we could go and live in a different EU country for a period of 3 or 6 months (I forget which), exercising my partner's right to free movement, and by virtue of that I would then be entitled to a Danish social security number (possibly with some annoying fees to be paid). Many people apparently move to Malmö for a while and continue working in Copenhagen to get around this, but it's a bit early to be considering that I think, and my partner really likes where we live (and waited a long time to get this apartment) and it would be a rather unpleasant commute for him from Malmö.
Ultimately, if I don't get a job by 8 August, I may be asked to leave. I've written to the British Embassy to see if they can tell me if this is likely, and how soon I can come back if it does. I will still want to do the Danish AS-levels in Chem and Physics as a self-teacher in the spring so I can apply to the University of Copenhagen in March, and I hope I can do this without a Danish social security number if I still can't/am not allowed to get a job. Then it's just a case of continuing to try to get a job when I can, and/or waiting to (hopefully) start the degree in September 2014.
That was long. But it's good to talk.