Is the floor of the toilet compartment like a shower tray? ie does it have a drain in a corner?
If so, I suspect that the mixer tap on the basin is not original and would have been a mixer with a shower head and hose that sat in a moulding where the current tap is and, when filling the basin, the hose dropped down into the basin pedestal. For showering the head was simply pulled out of the moulding and clipped into the bracket on the riser rail, the temperature and flow being controlled by the taps on the moulding on the basin. You'd have to reach through a gap in the shower curtain to do this - I notice that there is a shower curtain rail shown in the photo.
I expect a replacement mixer assembly could be bought from a dealer or you could try caktanks.co.uk