Having gone through the IVF process twice, once on the NHS and once privately, I can confirm that the private clinics use the same screw top lids. The private clinic doesn't expect you to use the disabled toilet and does provide jazz mags however.
The private clinic I went to had a room that clearly used to be a toilet, with a chaise lounge where the pan used to be, frosted glass window right next to the car park and a code on the wall for the WiFi, that didn't work.
Mine was the Lister Clinic in London. Room about the size of a box bedroom with a 2 seater sofa. Had a selection of very dated mags and a serving hatch like in a 70s hatch between kitchen and dining room, through which to serve your potted custard.
Having gone through the IVF process twice, once on the NHS and once privately, I can confirm that the private clinics use the same screw top lids. The private clinic doesn't expect you to use the disabled toilet and does provide jazz mags however.