I taught my DD (at her request) when she was 4 - 2 years before starting school here, as I didn't want her to be learning German phonics and English together, learning 2 different sets of phonics at once seems confusing.
I am just teaching Ds1 now and he is nearly 6 like your DD but I wanted to leave it til he himself was interested and not make it a chore. We are keeping him at KiGa an extra year anyway, so he won't start school til Sept, so again there won't be an overlap at the phonics learning stage, I would avoid teaching English phonics at the exact same time as French just to avoid confusion - do it either before or after.
We have used Jolly Phonics for the actual basic phonics (letter sounds and diagraphs - the sounds letters make in combination). It works well, especially the finger phonics books which you can use over and over with more children (or sell when you are finished) but we don't have any reading books that directly follow on. I also have work books and a teachers guide and wall posters - but that was slight over kill knowing I had 3 kids to teach and being a bit over enthusiastic with DD :) DS1 has not wanted to do the work books (his fine motor skills aren't great) but DD did the worksheets in the back of the teachers book. The Finger phonics books alone are enough to teach reading, the work books are more about writing.
For early phonics readers we have some early phonics readers from Usbourne - 'Fat Cat on a mat' and other stories (they are lift the flap and nicely illustrated and are more fun than the title sounds, but geared more at 4-5 year old UK reception age than nearly 6 year olds - they are pleasing though because children can read them using their phonics very early on). We also have a range called 'Usbourne Very First Reading' - a series of 15 consecutive books which come with a parents guide and use increasingly complex phonics, and then various other random Usbourne First Reading books (there are levels but we haven't stuck to them, just bought books that are of interest, mainly non fiction ones on elephants, bugs, space etc.) and we have quite a lot of Oxford Reading Tree books, partly because those are widly used in UK schools and I was thinking of awareness of Biff, Chip and Kipper being part of their "cultural capital" as British kids :o
Other people will tell you to just let her learn in French and she'll pick up the English on her own :o this is true of lots of kids, but I kn ow others here who don't read or write in English at all and don't want to - they will learn to at school obviously, but I guess it depends how much you care about allowing English to become a strong second language (despite being mother tongue) - I have a bit of a bee in my bonnet about keeping my kids' English skills as close to native as I can, and for me that includes being bi-literate as well as bilingual, if it can be done without causing anybody stress (if they really didn't want to I would drop it for that child) :o