I think you need to keep a diary and track all your spending for a month, as a lot of your figures are estimates. You can't really set a budget until you know exactly where all your money is going.
It doesn't seem as if you can do anything about your rent at the moment, and even if you moved somewhere cheaper, it's a lot of money to actually move.
You should be able to cut your groceries budget to £60 per week easily enough, and change your phone when your contract is up. It might be cheaper to have broadband at home and Giff Gaff phones, for instance.
I would also focus on getting your bills as low as possible. Go through everything and cut anything that isn't essential.
I would also think about trying to walk one of the pre - school runs, as walking there, back, there and back again is too much in a three hour session.
Finally, there are lots of us trying to live frugally over on the credit card forum. We are a friendly, supportive bunch, and you're very welcome to join us.