Like SD said that is exactly the same position as a self-employed person. I have family members who are self-employed and have mortgages.
I'm self-employed and have been for almost ten years and getting a mortgage shouldn't be a problem as long as you can present financials; most importantly sales, expenses, recurring revenue contracts and forecasts. Can a kid on zero hours do that? Highly doubtful. It's a disgrace!
Being self-employed and being on zero hours are two entirely different things. Self-employed means you are in control of your business affairs and can prove a trading history and your business has "goodwill" value. Zero hours employees could be fired tomorrow and your boss controls your hours. Difference scenario altogether. It should be stopped now!