Another answer suggested using the AC-volts setting of a multimeter, taking advantage of the DC-blocking filter it will use in AC-volts mode.
You could also build your own DC-blocking filter in front of an LED, e.g. a non-electrolytic capacitor and resistor in series with a pair of LEDs in opposite polarity. This is basically a probe that detects if any AC voltage is present.
+--->|------+
[oscillator]---||----^v^----| D1, D2 |---|
| C1 R1 +---|<------+ |
| |
+-----------------------------------------|
A one-diode rectifier would just charge the cap and get stuck with no current flowing; that's why you use 2 diodes in opposite polarities, so current can flow in either direction, through one or the other. (Only one of the diodes needs to be an LED.)
Choose a plenty-large capacitor (but it has to pass current both ways so it can't be electrolytic, and a resistor appropriate to not burn your LED if the full DC voltage was applied, and to not load your oscillator so much that it stops oscillating. (Each diode will actually only conduct about half the time.)
In case of very high frequency the capacitor will be basically a short circuit, but at low frequency it will have significant impedance. Still, with a large enough cap, you should still get enough current to make the LED visible. If your frequency is too high for the LEDs to really rectify, that's fine I think: they'll just both light.
You could build a bridge rectifier out of fast signal diodes as a power supply for a single LED (still with a cap in series before the rectifier as a DC-blocking filter, and standard filter cap + series resistor after the rectifier).
With the circuit diagram in the updated question, the LED connectsion seem strange to me.
You have the LED connected between the outputs of two op-amps with no series resistor. It will probably be lit at max brightness any time the voltage difference is above its forward voltage. (Surprised you didn't burn it out.)
You say you measured some DC volts and more AC volts with a voltmeter. That means you have oscillation, but also a DC bias. That DC bias might be enough to keep the LED permanently lit, so you don't see much variation from the AC voltage changes.
Other comments and answers have pointed out that you need to be careful that your measurement attempt doesn't break the oscillator itself: that usually means putting a decent-sized resistor in series with the LED if you want to connect it between points in your circuit that aren't like ground or an op-amp output.
Or power it from a high input impedance amplifier, e.g. using another op-amp as a voltage follower, if you want a high RC time constant in your oscillator but still need enough current to your LED.
But probably you can just connect it from an op-amp output to ground, with a series resistor, so you're not affecting the voltage across any of the higher-resistance resistors.