Recommendations for PCB manufacturers in England

It depends what you're looking for PCB Train is good for a handful of basic boards. For something more companies like Garner Osborne or Zot can be used. It depends on the complexity.

I need a first handful of prototypes: based on my layout I need the print, drill, solder and screen print, a 4 layer pcb with mostly SMD components. Not a complex circuit: power regulation, FRAM, Bluetooth module, two tactile buttons and indication leds, 2 terminal blocks.

As long as the minimum track and gap is not less than 0.125mm I would look at using PCB train (https://www.pcbtrain.co.uk/)

For a full set of their design rules / capabilities see here:
https://www.pcbtrain.co.uk/resources/pcb-technical-capability/

This is a low cost provider of PCB's and can make singles boards in really small quantities. If you need to scale up then you'll need ot find someone else but by the sound of things this won't be too hard it doesn't sound like there's anything scary on your boards.

I hope this helps

yes it helps.
thank you pcbmechanic.

https://www.maguselectronics.co.uk/ are a great company that I work with. They have a full design team for prototypes.