Wont work, cant work!
The inputs are effectively paralleled at the base of Q5 so it has no way of discriminating between either input active and both active. Also with the Vce_sat of Q1 and Q2 in series and at least 0.7V drop across R4, they cant get Q4's base voltage low enough to turn it off. The design is too broken to be easy to fix, and if not your own design, you should be very suspicious of any other circuits from the same source!
Try the attached circuit or sim it in
LTspice.
N.B. Vcc is the positive of your battery, and the input switches should be wired between that and InA, InB (without the pulse voltage sources). To probe logic levels without disturbing circuit operation, use another NPN transistor as an emitter follower to drive the LED. (i.e. signal to base, collector to Vcc, emitter to a 1K resistor, to the LED anode, cathode to ground)
Theory: Q3,Q4 perform a NOR function and Q5 inverts that to an OR. Q1, Q2 perform a NAND function which is wire-ORed
* with the Q5 OR to make the output low when both inputs are high, thus giving XOR.
*wire-ORing refers to the operation of combining open drain or open collector outputs if the logic is active low. For active high logic as in this circuit, the actual operation is AND.