Will an external pump raise water pressure enough to make a shower run hot?

If your hot water had hot water storage cylinder then increased flow rate with hot water will work. In case other hand a combi boiler depends on flow rate of hot water gallon per minute, if boiler is capable to give efficient hot flow it will work. A common boiler has result bw 50-70% improvement. Before installation check if you need positive gravity fed or negative feed pump.

Yes, if you run a small pump that will be enough and not to much pressure for the shower to handle. Salamander do great pumps.

There are some details missing from your description. But here's me having to assume? Triton Electric showers should be plumbed directly into the mains water supply. If your mains water supply is less than on bar that would be remarkable and a cause for concern! if your triton shower has been plumbed into a header tank that would be the problem with pressure.

You didn’t specify wether it’s a gravity fed system or not. But if it is poor mains pressure causing the problem you’re best bet is to fit a mains booster pump specifically designed for mains water (12 litres per minute max) this should cure the problem unless it’s being caused by restriction in the mains (crushed lead main etc)