I suggest contact HMRC in a way that simply expresses concern for getting your own tax affairs straight, rather than appearing to criticise your employer.
e.g: "I am currently employed by Arkwright's Store and, as tax isn't deducted directly from pay, not making appropriate payments to HMRC. I wish to remedy this matter but, as I understand it, I am not eligible for registration as a self-employed person. Please advise me as to the correct course for me to follow in this matter".
Keep a copy of your letter and then, if your employer complains that you've 'dropped him in it', you can show him that you were simply seeking to ensure that you stayed on the correct side of the law.