In that event, use two diodes on a power transistor's base.
... giving a wired OR instead of a NAND ... ?
are you thinking of a PNP transistor ?
The IR LED and resistor will be on the top transistor's collector. It's not uncommon to run an IR LED at 100ma or more.
For directly controlling such a load, a logic IC might not be an alternative, neither.
But the swing is definitely big enough to clearly switch a LED on/off.
BTW: 6V and 4.7k allow for about 1.3 mA, if I remember Ohm's law.
If you replace that 4.7k resistor by a 47 Ohm resistor and a IR led, you should also reduce the base resistors from 10k to something about 1k,
and check if the 2N2222 transistors won't get too warm.