My shower only stays hot for a minute and then goes cold even though the hot water works everywhere else. I have a hot and cold tap and a third one to direct to shower or tub.

your not making sense with you question --state it goes cold in middle of shower --but state it will stay hot or cold forever ----if its the latter continued hot till you shut off water that normal working if the unit shuts down during shower then you have a problem ----unit is set at one temp and runs at that temp and you use shower 1 handle / 2 valve what ever you own to adjust temp their are really only 4 reasons stops making hot water ----2 are usually well water supply --incoming water pressure /flow or to cold ----other are gas valve -or sensor so i would look into your shower handles since almost all have adjustments and anti scold safety

check tap had a shower myself that would start hot then 3 minutes later hot was cold! then hot after 5 minutes.
I turned off main tap on meter, Undid tap fitting on shower found the washer was worn but also the tap fitting was seized could not turn it for full movement. You could soak the tap screw fitting in CLR and pull apart and use a tap grease on the screw thread or just buy another tap fitting. I discovered most of my tap fittings had calcium build up in them. But I replaced the shower tap valve thingy as it was really seized up.

There is an adjustment knob to control how hot the water can become. There might be a problem at the tap (however, if this is the case, you would still get hot water everywhere in the house). If the entire house isn't getting hot water, and the knob is set at the hottest water, you probably need to get the hot water tank replaced.

Are the pressure ballance taps? If the shower is on and someone flushes the toilet the water will get hot in the shower,if they are not pressure ballance taps,or if they are the cartridge needs to be replaced in the tap.

Test for crossover, where bad faucet cartridge or recirculation check valve keeps hot from reaching taps
http://waterheatertimer.org/Crossover.html

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Possibly switch off hot and cold water then undo cold water flange closest to the outlet and see if water comes out.if it does, tap assembly is faulty. If not, go back to where next available connection to the pipe was made. Work back untill you find the problem.

Often, this is the result of a restriction of exhaust gases. This type of hot water heater must be adequately vented, (per installation specs). I found this out after taking a number of cold showers. :^( Two years later, and still counting, no more problems. :^)

there is a little cap on the hot and cold side above the shut off valves, turn the water off at these valve and take the caps off there are filters in there clean them off and reinstall them that should take care of the problem, you might want to flush the tank while your at it hook a hose to the hot side connector and run it out side or to a sump pump then turn on the cold water and flush it out you can also reverse this procedure by doing it on the cold side

If you are having trouble only at the one fixture (shower), and no where else, the problem is most likely in the shower valve...which either needs new cartridges, or a complete replacement.
hope this helps

hi did you run a seperate cold water supply from the storage tank to the shower pump & also a seperate hot supply from the hot water cylinder, it sounds as though the cold water is crossing into the hot service & pushing its way up to the tank as its the only way it can go, sounds as though the cold water connection to the shower is not tank fed & possibly directly off the water main.?