One small correction here - that three month "limit" is only if you want to transfer unemployment benefits from the UK to France. As an EU national, you have the right to live and look for work in France as long as you want to - though the general rule applies that, until you're contributing into the system (called "cotisations") you won't be entitled to benefits from the system.
Finding a job without any or much French is a whole different story, I'm afraid. The only real "English speaking only" jobs tend to be those reserved for monolingual executives transferred in by their multinational company. Some folks come over with the intention of teaching English, but the market for that is pretty nearly saturated. Even with a TEFL or TESOL qualification, the hourly rates are low and many language schools will only take on part-timers.
The other "practical" issue is that of finding a place to live, assuming that you'd be looking to rent. In France, you're generally expected to be able to provide 3 months' back payslips to establish your credit. So there's this chicken and the egg issue.
If you can stay with your girlfriend, that eliminates one problem - but be aware that you need to have private medical coverage if you aren't eligible for the national plan through work. It used to be possible to get an "interim" coverage by showing proof that you were covered in the UK up until your departure, but that option is currently available only for retirees. Your girlfriend may be able to add you to her national coverage - but only after you get PACSd or married.
So, in short, it's not all that difficult, but there are some specific hurdles to jump.
Cheers,
Bev