Hi,
I'm pretty sure the closest you're going to find is the 1055 service manual, I uploaded one to KO4BB a few years back...
http://www.ko4bb.com/getsimple/index.php?id=manuals&dir=06_Misc_Test_Equipment/DatronThe main difference with the 1059 is that it has a current converter board. The rest of it should be pretty much identical - although I think the 1059 has range LEDs, I'm not sure how those are implemented.
The cable from the Analogue board -often labelled DC isolator (Red Blue White) should be connected to the Volts/Ohms input terminals - Red:Hi, Blue: low, White:Guard.
I can't remember what current terminals the 1059 has. The brown and blue are clearly the current shunt connections, if the current terminals are brown and blue then they should match up one-to-one. The black wire appears to be also connected to current low (blue). This
might also go to input Lo but I can't be sure of this.
For now, I would just connect the voltage wires and leave the current converter wires safely disconnected (the current converter board is an option anyway). I can't see any sign of an ohms converter option board or an RMS (AC) converter.
You will hopefully find that the main Analogue board and A/D - Display board are more or less identical to the ones on the 1051 - Datron made a lot of use of common main boards and option boards across the 104x and 105x range.
Hopefully that will get you started (you're probably going to need some nuts for the terminals) and give you voltage readings.
If you want to post more photos then it should be possible to see exactly what capabilities it has (apart from 5-1/2 digit resolution DC volts and current). It would be interesting anyway, I've never seen the internals of a 1059 in the flesh.
EDIT: Just looking at photos from an old ebay listing, yes, Brown and Blue should go to the same coloured Current Hi and Lo terminals. I'm still not sure about the black, if it is indeed connected to the blue wire, then it probably goes to Volts/Ohms Lo. If separate then hold off.