I looked at the circuit quickly, I don't quite understand the circuit nor would I do it this way to get a constant current source. If you use resistors to simulate a current source, the output impedance is way too low unless you use very high value resistors which would limit you to very low current capability. Also, it only work if the current injection point is ground referenced. If the point is moving, all bets are off. It is much easier to make a current source with an op-amp and a MOSFET.
But be that as it may. To measure the noise, you terminate the output with a resistor and measure the noise. You can use a RF noise meter for measurement. It is possible to measure with the scope, but it is not accurate because it is hard to determine the exact amplitude. I did a lot of noise measurement with scope just because I did not have the noise meter. I had to make judgement what is the amplitude as all you see is a fuzzy line on the screen!
The difficult part is you likely have to put amplifier stages to amplify the signal before you can use the noise meter and the the amplifier themselves produce noise. With the noise meter, it's easier, you put a resistor at the input of the amplifier chain, measure the noise. Then connect the current source and see the increase of the noise. Then you do calculation and get the noise contribution of the current source alone. Remember, noise are root mean square, not direct addition.