Waiting at a tap for hot water to appear is a common complaint, and it is usually not a fault with the hot water system at all.
What comes out first is the water already sitting in the pipe between the unit and the tap, which has cooled since it was last used. The further the outlet is from the hot water system, the more cooled water there is to clear before hot water arrives. That volume goes down the drain every time.
Uninsulated pipework makes it worse, because water in a bare pipe loses heat faster. Lagging, which is the insulation fitted around hot water pipes, slows that loss. It does not make hot water travel faster, but it keeps the water in the line warmer for longer between uses, which shortens the wait when the pipe has been idle for a while.
Pipe run length is the other factor, and it is set at installation. A hot water unit positioned a long way from the bathroom will always have a delay, which is worth considering if a unit is being relocated or replaced anyway.
If a hot water system is leaking, making unusual noises, or delivering water hot enough to scald, switch it off at the isolation point if you can do so safely and call a licensed plumber. Tempering valves and pressure relief valves are not do it yourself items, and a failed relief valve can be dangerous.
