I agree it would be best to take it all off, and the reason why is that if you redo the job, then the new primer and paint will be better suited to the service conditions and is likely to provide better service. You don't want to paint over paint that's not adhering well, and have the new paint sticking tenaciously to the old paint as it's peeling off.
A lot depends on whether or not you have shower water impinging on these painted tiles. If not, I'd proceed as follows:
1. I'd prime with Zinsser's Bullseye 123 water based primer. Zinsser advertises that this sticky primer will adhere well to smooth surfaces including glazed ceramic wall tiles. Zampa tells me that Zinsser's products are available in the UK, but you might have to phone around to see who sells them.
2. Part of the reason why the old paint may have come off is simply because not all paints are equally resistant to moisture. Some paints, particularily those made from polyvinyl acetate (PVA) resins have a tendancy to soften up considerably and loose their adhesion to the substrate when they get wet. So, as a top coat, I'd use a Kitchen & Bathroom Paint, or a paint made specifically for bathrooms. These Kitchen & Bathroom paints or Bathroom paints will be made using a Perspex resin that is particularily good at retaining it's hardness and good adhsion to the substrate even if it gets wet.
If you have a shower spray, then you can TRY getting away with water based primer and paints, but I think in a shower you're eventually going to have even this waterbased gameplan starting to peel off. In that case, I think you're best bet would be to either roughen the surface of the glazed wall tiles and paint with a polyurethane based paint (or even an "oil based") paint, or use a two part epoxy paint on the tiling.
I've never painted with epoxy paints, and I've never painted glazed wall tiles, so I can't really advise you on doing this.
I'd also phone around to any marinas or boat stores in your area and see if you can find any paints made for painting boat hulls. They would also be very water resistant.