After replacing R4 and R5 on the top PCB assembly, the voltages at the +6V and -25V outputs were restored.
The +25V output was still 0V and could not be adjusted. Previously I checked the input and output voltages on the U10 and U9 opamps (TL074CD in voltage follower configuration). One of them, U10D had different voltages at its input and output (-2.2V and 14.4V). After replacing U10, all 3 outputs are working with the unreg annunciator on for the +6V ouput.
The pics before and after the repairs are presented here:
So after initial repairs on a broken E3631A, I still have the "unreg" annunciator on when using the +6V output, however all the outputs are now working properly after changing the R4 and R5 resistors and the U10 opamp. Similar problems were discussed on eevblog, which inspired me during the repair. This problem ("unreg" on) is due apparently to a faulty LM339M or HP 2211 (U1, U2, U3, U4...). In this short video I show that the opto-isolator U3 on the bottom pcb assembly is faulty. Unreg is lit only when the PSU is working in CV mode. In CC mode there is no annunciator displayed, which explains why only the U3 is broken (see the schematic in the service manual: Agilent_E3631_Power_Supply_Service_Guide.pdf).
I believe LM339M is also broken, so I shall also order that one, just in case...
Update: the HCPL 2211 is replaced and the unreg annunciator is not on any more.
Yet the last digit changes when I try ot modify the higher digits and I believe to fix that problem U9 needs to be replaced too.