Hi Mirko,
first, the good news is that the names recorded in PLBORN (BHPS) and PLBORNUK (UKHLS) refer to places that can, in principle, be linked to official geographies. The ONS geography portal provides numerous shapefiles; it is not something we provide. Please visit http://geoportal.statistics.gov.uk/
As to the content of the variables we provide, the code frames for the BHPS and UKHLS are slightly different. In Understanding Society there is a coding frame and the administrative area is county. The list is historical, so it includes counties that no longer exist. We attached the rules file, which is used to code the question. The BHPS was slightly different, the question itself asks for place of birth town and county. These were then coded by the fieldwork agency (NOP) who would have used a similar strategy as the one outlined in the attached document.
In order to link these names of places/counties to geographies for which ONS provides shapefiles you will need to work out which county the places were in during the reference year you chose (based on the place names; the values we provide do not refer to official geographical identifiers!).
PLBORNUK is included in the standard EUL version of the data. The Special Licence version of the data also includes PLBORNUK_ALL which will provide additional valid responses, i.e. for cases that provided an uncodeable response.
I would recommend that you try and work with the standard EUL first so you can see whether your method works and only apply for the Special License if there is proof of concept.
Hope this helps,
Gundi