Replace two way light switch

https://www.homeimprovementweb.com/information/how-to/three-way-switch.htm

switch out the light with a known good blub
maybe the lights are bad

--------------- wire #1 ---------------------
power in --- -----power out to
--------------- wire #2 --------------------- light

the upstairs switch is improperly wired. With the breaker off, un wire the upstairs switch. Using an inductive tester with the breaker back on (use caution not to get shocked, determine which wires get power from the downstairs switch in EACH position (there is no 'off' in a three way). when you determine which two wires bring power to the upstairs switch, the remaining wire is the light.

Hi Lyn,

Had the same problem with my system. turned out to be the thermostat in the hallway. Mine is an older electro-mechanical type with contacts that open and close as the temperature changes. (these type of thermostats click as you rotate the temperature dial) What had happened was the contacts had stuck closed even as the temperature rose and thus the boiler kept going. I cleaned the contacts with WD40 oil and everything is fine again. If you have the same type of thermostat this may be the problem. Be aware that if you or anyone else removes the cover of these thermostats there is live mains at the faulty contacts, be sure to switch off the circuit feeding this thermostat or remove/switch off the fuse in the main fuse box. Alternately you could replace the thermostat. If you have an electronic digital thermostat then this solution is not valid.

Good luck

replace with orignial stile switch

Is the bathroom light controlled by two switches?
Is the hallway light controlled by two switches?
What color wires are in the bathroom light switch that you changed?
Side wiring will not change the way the switch works.

I don't know the copper-brand 1301 switch
Internet photos of the 1301 show that it has two screw terminals, and a green ground screw
Switch with 2 screws is a single-pole switch. (plug green ground screw)

Sounds like you are replacing 3-way switch with 3 screws. (plus green ground screw)
3-way switches control hallway lights, with switch at each end of hallway.

I think you need to buy a 3-way switch.

How to wire 3-way switch:
Bare ground always goes to green screw
The key is the dark-colored screw.
Old switch has dark-colored screw >> mark this wire.
Attach this wire to dark-colored screw on new switch.
Other two wires go to either screw.

Replace batteries

What it sounds like to me is that the second light may have fluorescent bulbs - either the compact fluorescent type or maybe a circular bulb. The fact that when you turn off the switch and then immediately on and it works fine, also makes sense if that is the case.

Im not sure if you know what you got yourself into but i hope this helps. Sorry for all the words but this does take some reading if this is new to you.
http://www.homeimprovementweb.com/information/how-to/three-way-switch.htm