Okay, now I'm not sure what's going on. The time constant that I solve for is nowhere close to the time that it takes for the LED to come on.

That is, if I use a 330kΩ resistor and 470μF capacitor, I'm expecting about 2.5 minutes, but the circuit really ends up taking more than 5 minutes.

Does this have to do with the exponential rise somehow getting slower as Vref is reached?

Also, my circuit looks exactly like the one in the picture I've attached. The only difference is that R5 is 330kΩ

 

When a circuit doesn't behave as expected, you want to find out what part(s) of the circuit are deviating.

It could be that you are losing charge on the capacitor because of its effective shunt resistance. Measure (or record on a scope, if it's an option) the voltage on the capacitor with a good voltmeter (input resistance in the 10Mohm or more range) and record how lont it takes to get to 10%, 20%, and so on of the supply voltage. Also, measure the voltage of the voltage divider that goes to the inverting input of the opamp.

 

I told you earlier why Tantalum caps are preferred for timing circuits. Large electrolytics are leaky. That said, you can lower your reference voltage by changing the value of the 100K. Doing so will lower the trip voltage, which will shorten the time out.