You should only register as self-employed if you feel you are genuinely self-employed. Given that you don't work for more than one "employer", you are an employee as far as I'm concerned. The people who are running the business are heading for trouble if this is how they operate.
You really only have three options.
1 Confront the company and ask why they are not operating PAYE. Clearly, they want to avoid employer national insurance contributions which are a high burden on employers. If you do confront them, you may be seen as a troublemaker. If the company wants to treat you and others as self-employed, they should at least tell you what to do as far as your tax situation is concerned.
2 Contact HMRC anonymously and tell them the situation. Avoid giving your name. Put aside the tax and NIC in case you and not your employer is forced to hand it over after a PAYE audit by HMRC.
3 Register as self-employed and hope that the situation doesn't come to light. The employer could get into trouble if they have not enrolled you into a pension plan or are not contributing to an existing one set up by the previous business owner.
4 Do nothing, (not a sensible option).
I'd be inclined to make the anonymous call to HMRC. However, such a move and the following PAYE audit may push the company to close the part of the business you work for.
I hope this helps but let me know if you have any further questions.