ABS system fault with code C1063

Good to know you have got to the bottom of the defect. I would recommend a complete fluid change. The removing of the pump unit should be just a spannering job and should not be too difficult. The bleeding can be done piecemeal as in start at the furthest wheel from the reservoir and open the cylinders bleed nipple at that wheel and get somebody to pump the pedal. Don't let the reservoir run dry though by adding the new fluid. Keep doing that whilst going wheel to wheel furthest to nearest and see how it goes. It might help if you can blank the disconnected pipes at the hydraulic unit as you disconnect them to try and keep the pipes full of oil rather than air.

Thank you very much for the suggestions on how to bleed the brake fluid.

A note to other readers about the electrical part of the ABS pump:
It is a bit tricky to properly solder the connector pins and other electrical elements on pump's electrical board, so be careful which electrician you choose to do that job (if you happen to have the same corrosion issue).

The mechanic employed a similar bleeding procedure as the one which you described, except that (I think) he opened all three bleed nipples at the wheels simultaneously and let the brake fluid bleed in parallel on three wheels. There was no bleed nipple at the rear right wheel - the brake piping just passes through the brake cylinder in that wheel (pass-through point).
The mechanic did not allow the brake fluid reservoir to empty completely during the flush - he periodically topped it up during flushing. The draining was stopped and all bleed nipples closed when clear new brake fluid started to bleed from all three wheels. After that no air purging was necessary.

Now my ABS pump is working normally and the brake pedal is OK, no air in the system.