Sounds like an issue with your heater. Not sure why your dishwasher changes anything, they normally run off cold water and use their own heaters. Sounds more like you run your kitchen tap while filling the dishwasher, and flushes the cold water out of your pipes, and gives your heater enough time to get to temp.

Have you tried see if you can set the output temperature of your heater higher? Could be the output temp of the heater is very close to “shower” temperature, and a lot of heat is lost heating up pipes etc, something that could be sped up by increasing the heaters output temp, or improving the insulation on your hot water pipes.