I've been lucky to find several similar circuits on the web, as well as datasheets. Here's a list for those who are as curious as me...

• http://apollo181.wixsite.com/apollo181/programming-74188-ttl-prom

• I once found a page but it is now closed. Fortunately I saved the page "just in case" and here is the programmer's circuit:

I'm not sure to understand most of it but it contains some interesting ideas.

• http://www.retrotechnology.com/dri/prom_shook.html and http://www.retrotechnology.com/restore/fuse_bipolar_prom.html
  

Note: this schematic applies to the 82S23 which has a particular, slower programming cycle, with a 1s pulse, limited to about 65mA (hence the series resistor and the push button), as found at http://www.avrfreaks.net/forum/scheme-prom-ic-programmer :

Yup, Vpp is also applied to the data pin, not Vcc...

Note that current-limiting the Vpp node should be possible with a suitable base resistance...

• http://www.seekic.com/circuit_diagram/Basic_Circuit/PROGRAMMING_OF_PROM.html

• http://www.seekic.com/circuit_diagram/Amplifier_Circuit/PROM_BURNER.html (but for the similar 82(S)23 where Vpp is applied to the data pin, not just on Vcc)

• http://www.abicko.cz/clanek/serialy/5964/cislicova-technika-87.html (not very useful, unless something is hidden in the text which I don't understand)
• http://www.iva-w.org/ZXS/sinclair_zx_spectrum.htm : broken site ? Fortunately I saved the schematic :

• Another one (source forgotten, maybe from one of the above broken links ?)

• Another set of pictures I had saved but couldn't trace to the origin. I appreciate the (quite clumsy) translations :-)

(ok it's obviously a translation and redrawing of the original TI datasheet but it's still nice)

• PROMs from National Semiconductors are described in section 5 of http://bitsavers.trailing-edge.com/pdf/national/_dataBooks/1977_National_Memory_Databook.pdf
  There is also the 1980 version at http://mirrors.acm.jhu.edu/bitsavers/pdf/national/_dataBooks/1980_Memory_Data_Book.pdf with the programming procedure at page 6-28 with a slightly better description