Why does hot water only run for 30 seconds in bathtub?
You are pretty hardy if you can take a cold shower in NY in the winter or spring . . . or even summer. We have the exact same old fashioned tub/shower with old Price Pfister separate hot and cold valves with a diverter in the wall on a line between the hot and cold handles. I have taken this type of tub/shower valve apart and they are pretty simple. I do not see how any problem with the diverter could cause cold water to come from the hot water line. What do you get if you turn on the cold water valve and leave the hot valve shut?

Are the cold and hot pipes connected by an anti-scald device?

Some models of anti-scald mixing valve are designed to be placed at the output of the boiler, but others could be just upstream of the shower. The fact that the lavatory delivers hot water continuously as expected but the shower does not, could be due to the presence of a hidden mixing valve in the wall behind the shower. The scenario would be that the hot water is so hot that it drives the valve to the cold stop.

Try turning on the shower cold water valve first to a moderate or high flow and then metering in increasingly larger amounts of hot water to get to warm. Maybe a relatively high flow rate of cold water from the beginning will prevent the thermostatic mixing valve (if there is one) from being over heated and over driven to the cold stop.