I would get rid of the -15V supply and select an appropriate opamp. An LM358 would kinda work, but it doesn't have nulling offset pins (if that matters, since you'd probably get only a few mV offset), and whereas its output can get down to almost ground, it won't get high enough - due to the current it will have to sink - to get your full output voltage range here IMO.
You've got that backwards. The LM358 won't do without a negative supply because it can't sink enough current with a low enough saturation voltage for 1.25V out. The higher end of the voltage range is no problem.
Well, yeah. The sink current (as opposed to source) is going to have the opposite effect. My bad.
That said, I think an LM358 can sink 5mA without much problem. Its output should be able to get as low as a few tens of mV, and I think it will still get further away than this from the positive rail on the other end. From its datasheet, it's given for a typical 8mA of max current sink. Would have to be tested, but simulation shows that the output can go as low as ~30mV for 5mA of current sink, and can't get higher than the positive rail minus ~1.2V, so there's still this asymetry (of course for higher currents, the output range would get shifted up). Looking at an LM317 datasheet, its reference voltage is specified between 1.2V and 1.3V, so a minimum of 50mV output for the opamp is not really going to make much of a difference actually. If the OP really wants 1.25V min voltage output to a few mV of accuracy, then clearly this topology is not for them. Given the proposed schematic and their questions, I doubt 50mV or 100mV from 1.25V will make any difference! But if it did, I guess an LM317 could go lower than its reference voltage by setting a negative voltage on its ADJ pin, but not 100% sure it would not cause issues. In this case, a slightly negative supply voltage (-15V would certainly not be needed) should allow to do this.
(As to the higher end, there is a lot of headroom here actually, if the opamp is powered by the input voltage of the circuit, given that the LM317 has around 2.5V of dropout voltage and there is an extra 1.25V, so that's 3.75V of headroom. So yes the higher end shouldn't be a concern here anyway.)
The OP's schematic's idea was actually given here:
https://www.edn.com/design/analog/4363990/Control-an-LM317T-with-a-PWM-signalThere are of course a lot of ways to do better, but as far as I've gotten it, the OP simply wanted something that "works" before going any further and chasing after a few mV...
![Roll Eyes ::)](https://www.eevblog.com/forum/Smileys/default/rolleyes.gif.pagespeed.ce.XcUK202Yqt.gif)
And in that case, it seems to just be a matter of using a more appropriate opamp than the TL081.