Yes, you can apply for One Parent Family Payment, and you should do so as soon as possible, as late claims aren't always backdated.
You will have to show some evidence of the amount being paid, SW will tell you what they need / what's acceptable to them.
If you have no other income while you're waiting for One Parent Family, you can contact the Community Welfare Officer at your local health centre and apply for Supplementary Welfare Allowance, which can be paid until the OPFP comes through.