My first guess would be that you don't qualify for a visa to join her - unless you are married or have been living together for at least 2 years (there's a spouse dependant visa, but no girlfriend/boyfriend dependant visa).
If you plan on going back and forth between the US and the UK, you should be able to apply for a multiple entry visitor visa. I'm not sure how that would work in terms of the 6 months out of 12 rule, though. It's about 430AM in the UK right now, so give it a few hours for the UK folk to wake up.
Becca suggested a visitor visa, but I don't know how feasible it would be, because every time you returned to the UK you might be at risk of being refused entry if the immigration officer believed you had already spent too much time in the UK (you're only supposed to spend a maximum of 6 months out of every 12 months in the UK as a visitor). Also, you're not supposed to be 'living' in the UK as a visitor and it would be illegal for you to carry out any kind of work while in the UK - you mention continuing to work for your US company, but you would only be able to do this if you only carried out the work while in the US - you can't do any long distance 'working from home' while in the UK as a visitor.
It's worth looking into whether it's at all possible for your EU citizenship to be processed faster than 12 months so that you won't have to worry about visas
.