The GoL boards probably want more current than the transistor can supply.
For the sake of this discussion, you can think of a BJT as current amplifier. A small amount of current flowing into the base controls a larger amount of current flowing into the collector. The exact ratio varies, but 100:1 is a good rule of thumb these days.
It looks like you have a 47k resistor sending current from the 3.7v rail to the transistor's base, with the photoresistor providing another path to GND. The photoresistor's resistance falls when it's exposed to light, so it steals current from the transistor's base, keeping the transistor shut off. When it gets dark, the phototransistor's resistance rises, and eventually most of the current flowing through the 47k resistor goes into the transistor's base.
The base voltage will be about 0.6v when that happens, so you'll have about 3.1v across the 47k resistor. That means about 66 microamps will flow into the base.
If we multiply 66uA by 100, we get about 6.6mA. That's enough to light an LED, but not enough for an array of GoL boards.
You can solve that by doing the multiplication thing again:
Sending the current from the first transistor into another one's base gives you another factor of 100 in current gain.
You lose a little current through the 47k resistor because both transistor's bases need to be about 0.6v, but the current gain of 10,000 more than makes up the difference.
That circuit does have one weakness though: the current will change gradually as the photoresistor's resistance changes. You can see the same effect in the LED setup you have. The LED fades in and out as the light changes. The circuit above will do the same thing with a hundred times as much current.
You can improve things by putting a PNP transistor in the middle:
A hundred times as much current will flow through the PNP's collector as through its base, so the voltage across the 330 ohm resistor between the PNP and the PN2222 will change about a hundred times as quickly as the voltage across the 330 ohm resistor tied to the PNP's base. As a result, the PN2222 will turn on and off much more quickly than the output transistor in the previous circuit.
The PN2222 is an NPN transistor by the way, despite its name. I've called it out specifically because it can handle up to 1A through its collector.
I'd personally make one more change to the circuit though:
The 100k resistor between the PNP's collector and the sensor input creates a weak positive feedback loop, and gives the circuit a property called 'hysteresis'.
The first transistor turns on when the voltage between the 47k resistor and the photoresistor rises above about 0.6v. When that happens, the voltage at the top of the 330 ohm resistor tied to the PNP's collector also rises. When that happens, the 100k resistor sends even more current into the first transistor's base, turning it on even harder. That raises the voltage at the PNP's collector even more, sending even more current through the 100k to the input, and so on.
The process forces a fast turn-on as soon as the input rises above the level necessary to get the loop started.
It does something else though.
When the first transistor is turned off, the 100k resistor is more or less in parallel with the photoresistor (through the 330 and 10k resistors to GND). That pulls the voltage at the bottom of the 47k resistor a bit lower. When the first transistor turns on and the positive feedback loop takes over, the 100k resistor is more or less in parallel with the 47k (through the PNP's collector to VCC). That pulls the voltage at the bottom of the 47k resistor a bit higher.
That voltage gap between 'pulling a little lower' and 'pulling a little higher' makes the trigger less sensitive to noise near the turn-on/turn-off level.
Putting numbers on it, the voltage between the 47k resistor and the photoresistor will reach 0.6v when the photoresistor's value is about 9k. If we put the 100k resistor in parallel with that, the photoresistor's value can go a bit higher (to 9.9k) before the transistor turns on.
When the circuit turns on and puts the 100k in parallel with the 47k, the parallel resistance is about 32k. To pull the voltage at the midpoint down below 0.6v, the photoresistor's value has to fall to about 6.2k.
In other words, the photoresistor's value has to rise above 9.9k to turn the circuit on, but then has to fall below 6.2k to turn it off. Once the circuit turns off, the photoresistor has to rise above 9.9k to turn it on again. The on/off values aren't at the same point any more, so the circuit can't flicker back and forth if the photoresistor hovers between 8.99k and 9.01k.
That separation of on/off values is called 'hysteresis'.