Are you saying that it is leaking from where you disconnected the hot water system? The taps that isolate hot water systems rarely work properly. Replace the tap or cap it off. Those taps make it a prick of job when I have to replace elements!

this means that one or more of your hot water faucets are leaking cold water through them.. make sure they are all turned completely off..
this occurs because there is no pressure in the hot water lines. if a hot water valve is not completely seated then cold water will run through your spout and also into the hot water line since there is nothing from keeping it from doing so.. you have to cap your hot water line.. or you might try shutting the hot water valves under your sinks.

If you just want to get the cold on, add a valve to the open end hot line. or just cap it. If you valve it you wont have to turn the water off to install the water heater. The cold water is making its way into the hot at your single handled faucets or shower.

All the water comes in from the cold side so if you open a hot water faucet cold water will come out the hot side. However if you turn the cold water valve off at the tankless coil (feed side )this should remedy your problem. If not the valve is bad and you should install a new one

its mixing through a valve somewhere
in this situation its normally a single handle shower valve that causes it. Shut the main off and you’ll be fine

There’s no heat.