I don't know why people keep suggesting that the OP's ex get a childminder, when she's said at least twice that she isn't happy about him doing this.
OP - you can do one of two things - remind your ex firmly that it is his responsibility to have the DC as agreed, and then be OK about him using a childminder or after school club. Or accept that you're choosing to take on the role of single/chief parent where you call the shots but also pick up all the responsibility if your ex declines.
But thinking more creativily, would there be a different contact pattern that would fit with his work and yours? What if he dropped them at your mum's after having them for dinner, or with her to take to school first thing?
Or could you suggest meeting with you ex and his DP to chat about the DC',s behaviour, swap ideas on how to deal with it, and see if she can be persuaded to pick up from school with a bit more support? I used 1p fines with DD when she was at a difficult age, which worked brilliantly, and allowing someone else the authority to do something like that (deducted from pocket money) gives them clear authority.
It can be really hard as a step mum to know how to handle a child who's misbehaving. As her own child is much younger she won't have much experience with your DS's age, she may be uncertain what sanctions she can enforce for bad behaviour, and she won't feel as secure in her own relationship with her DSC as you do with your own child. I can get as strict as I like with my own and still know they love me, but I don't feel that secure with my DSC, so it's much harder to be strict.
Also worth asking the kids what they want. If they're already doing overnights at your mum's because of your shift pattern, they may feel they're quite happy being at home with you a bit more and just going to their dad's on a Saturday morning. If you'd actually prefer them to be with you than with a childminder, then maybe the new arrangement isn't suiting everyone too badly?