How do I build a one shot non-retriggerable pulse generation circuit with 2 triggers?
A PIC 10F200 can do this job easily.  Since this is working on a human time scale, the microcontroller will react instantaneously.  Humans don't notice a delay up to a few ms or low 10s of ms, so the PIC 10F200 running at 1 MIPS is plenty fast enough to poll the two inputs, decide what to do, and light or unlight the LEDs accordingly.  Even if it took 100 instruction cycles to react (that's a lot), the delay would only be 100 &micro;s.  Nobody will ever notice that delay.

Another thing you should be aware of is that switches bounce.  You need to add deliberate debouncing code so that the "chatter" when the switch closes and opens doesn't get interpreted as a lot of fast button presses.  This is normal and common thing to do with mechanical switch signals.  Look up "switch debouncing" for more information.