In the US, you need to run hot water before starting the machine, but once the machine is running there is no reason you shouldn’t be able to use the sink as normal.

The federal government put restrictions on how much energy a dishwasher can use in a cycle, which has significantly restricted how much time they can spend on heating water and drying the dishes at the end of a cycle. In order for dish detergent to work properly, the water needs to be at least 135ºF, most machines aim to reach 140ºF. Most manufacturers will also state incoming water needs to be 115-120º minimum in order for them to be able to work (which is a common-high setting for a hot water heater - 120ºF is hot enough to scald you if you are not careful). If you do not get the cold water out of the pipes between the hot water heater and the kitchen, then the water starting temperature won’t be high enough and the dishwasher won’t work as well (specifically the detergent won’t work as well).