although we can, allegedly, travel within the european union without restrictions, the uk have maintained border restrictions so you wont have any problems getting into france, its the coming back that might prove difficult. be safe get a passport, i had to do mine and my childrens in a day recently and the passport office were very helpful, all done by appointments now so you dont have to queue for a week etc, ring em.
You pick up an application form at the main Post Office fill it out with photographs which must be counter signed by a responsible person ie Doctor someone thats known you over two years no relations. You should get it back within two weeks try too avoid peak seasons