Things are different, to say the least, over here, even in rural areas that don't have their own gas supply to run a boiler and pull their own water from a well. But at the end of the day, heating water is heating water.

Ignoring everything else for the moment, you mention a capacity issue which sound like it's boiler-specific, not pressure-related. In that case I'd start first by replacing the boiler if it's feasible, with a more powerful unit. I'd think once that's done you'd be able to install whatever plumbing you'd like to get you both hot and cold supplies at your shower.

This is the kind of boiler that fires up only when water is flowing through it right? AKA, "on-demand water heater" or "tankless water heater" here in NA, such as those produced by Bosch, Takagi, etc.

BTW, in a very quick search I've found power showers (a pressurized shower system) that works with both your hot and cold supply. But it's suitable only for a low-pressure supply.

The municipal water supply pressure around these parts is relatively high. When I lived on a rural property with our own supply we had a large pressure tank to serve the entire house, so every plumbing fixture in the place, including showers were quite standard affairs. It seems like picking fixtures is a lot more complicated in the UK.