Just to pick up a couple of points.
So far as I know there are no rules requiring "time and a half" or any other additional payment to work overtime.
Keeping the P46 certainly wouldn't prevent the employer faking a signature Brenda, it's just a blank document of which they will have dozens of them most likely. If they want to fake it they could just write another one.
It would appear the "employer" here has paid the girl "off the books" ie without officially employing her. That's only of any real significance if her earnings were over £110pw though I expect they were with those sort of hours. If they were over that then she's certainly underpaid NIC and probably tax though that depends on other info we don't have.
The Revenue say they haven't heard of the the "company" (I'd guess it isn't a company in terms of being a Limited Company, it will be a sole trader most likely). I would expect that means they haven't heard of them as being an employer rather than that they haven't heard of them at all.