Bypass connects cold inlet (bottom valve/lie) with the hot outlet (top valve/line) that is why you can get cold water flow thru hot water side.
Normally with a 3 valve system on water heater no check valves are used.....normally.
Cold inlet allows water into WH Tank......valve should be OPEN
Hot outlet allows water from WH Tank......valve should be OPEN
Bypass...should be closed (Open when bypassing WH Tank. Close cold/hot valves)
Something is plugged up.
Most likely the hot water outlet/valve
Could be cold inlet....easy to check by opening the Pressure Relief Valve on WH Tank (CAUTION..Water could be very hot and will splash)
If you have water in the tank (flows out PRV) then you will need to disconnect hot water line at water heater and check for plugged nipple or if hot outlet valve is not opening. (Turn off water supply and have a bucket/towels ready)