To get on a PGCE or School Direct course you will need to be able to teach at least one other language (I means as well as French), and it will have to be one that is mainstream in state schools - eg Spanish or German. There are bursaries available, but you have to pay university fees of up to £9k for the course. Your English has to be good enough to write academic essay. If you are a native speaker and don't have an additional language to offer, you might be best applying to an academy or independent school as an unqualified teacher. State schools are moving towards predominantly Spanish as it is easier the French and so better to bump up grades, whereas Indies still favour French.
I was in similar position ten years ago (feeling old now).
I have a degree from Germany and am native German.
I got my QTS via the Overseas Trained Teacher Programme. When you google it you'll find many universities offer it.
Basically, it's like the GTP, which means you get your qualification while working at a school. It is hard work as you need to do all the paperwork and collect evidence yourself, but worth it. You work, gain experience and get paid, what's not to like? All you have to do is find a school that keeps you on for about six months, it can be on a supply basis. This is what I did and then straight after I got my first permanent job.
You may find an independent school that will employ you as a teacher while you do your PGCE in your own time (if you can afford it you could work part time). It´s mostly reading up, then writing essays, and the occasional weekend of seminars at the university - I believe Nottingham is one that offers this.
The school would probably want you to have some teaching experience (doesn´t have to be French, or to kids) and / or general experience with kids, but really it would be up to you to decide whether you could plunge into the deep end.
Someone I know did a similar thing in a state school - worked a part-time timetable and had an experienced member of staff as a mentor then did essays etc. in his own time and gained a PGCE at the end of the year. However I´m afraid I don´t know any more details.
Whilst ideally yes, you should have a second language if only to open your career location options, many PGCE providers will accept just French. My PGCE cohort eight years ago was 50% monolinguists (myself among them). Most of these were native speakers of French. It does limit your options as most schools want at least two languages (even if one is only to KS3 or KS4) but this is slowly changing as schools adjust their timetables to accommodate more science/maths/English and less of everything else. In the four schools I have taught in two offered just French.