My daughter attends a state school with a very strict uniform

There is no national law on this. Each local authority has a uniform policy and each individual school has to interpret and apply that policy to the school. You should ask the school/local authority for a copy of its uniform policy. I don't know where you stay but I see that the Borders Council school uniform policy is freely available online and I don't think the other local authorities will be terribly different. It has to be said that the implementation of the policies vary widely from school to school. Some schools interpret the uniform as having a tie or a badge and anything else is up to the parents or the child. Other schools have a strict, "smart uniform" policy. However, the schools have to observe equality legislation and the argument would be that girls should be allowed to wear trousers (and that boys should be allowed to wear skirts). This something that has not been tested in the courts as far as I am aware but I think you have a strong argument that not allowing your daughter to wear trousers would constitute discrimination on the ground of gender. I think that you might want to seek a meeting with the head teacher and thereafter if necessary with the local councillor or education conveyor on the council. Happy to discuss further.