It is my belief that it is an act of kindness to teach a child to read. If your child is in the French immersion or bilingual programs, you may find that reading instruction is delayed until oral fluence is increased, possibly till grade 3.
If that delay for both English and French skills of reading frustrates your child, or you, you may wish to on your own supplement the school instruction. The child, in my experience can easily handle what you do at home, gently and gradually, without a real problem of conflicting messages since the two languages are similar in alphabet, in basic skills left to right, and in the sounds of most letters. The exceptions are ones the child will easily adjust to. So I would encourage you to teach the child to read English or French as you wish.
If you would like to keep up and enhance the French skills and I applaud you for doing this, the time you and your child spend together will be a bonding time and will review and enhance skills for both of you. It is very easy to do one page a day of the activity book, ensuring the child then has been exposed to basic reading skills in a logical way and enabling the child after even book one to read several hundred words. If you took any French in school yourself, you will find that the instructions are adequate for you to proceed through the course together. Each page only contains a few words and pictures, with suggested songs to sing or games to play. If you keep a dictionary handy for the odd time you are not sure what a word means, you will do fine.
If you choose to let the school handle French instruction while you focus on English instruction, that system also works well. The reading skills you teach in English at home will serve your child well and will transfer well to the French system also. My reading course Anchors and Sails may well suit your needs.
If your child is struggling with the bilingual program or unhappy there, you may feel very passionate about helping and this is one way you can help. Often it is the very clever children who are most frustrated when a system does not seem logical to them.
Showing them the logic of reading French (or English) seems to give them a whole new lease on life.