If the timer interrupt is not working at all, then either the expected event is not happening, you haven't enabled it or some other higher priority interrupt is stuck in a loop. If it's working but is executed after UART interrupt, it doesn't impact the measured data because the time value is captured by timer hardware independently.
Additional tip - use timer not only for input capture, but also use an another channel of the same timer for trigger signal generation. That way the whole trigger-capture process can be done by hardware and you will get the best accuracy unaffected by software timings.
