As no answer has yet been attempted, I will give the disappointing results of my efforts to find this information - but still live in hope that someone can come up with a better solution!
The bottom line is that information about sub-district extent does not seem to be publically accessible.
The Office for National Statistics provide a number of resources about administrative geography on a page titled Names, codes and lookups.
I had hoped that the sub-district codes would be included in the Standard Names and Codes (SNAC) database, which although it contains a vast number of administrative codes including those for registration districts, does not include sub-districts. The SNAC was produced annually up until 2011, but previous versions cannot be located on their current website. Previous versions of the SNAC can be downloaded from archived copies of the website.
The Statistical Geographies Database, a spreadsheet available for download from the Open Geography Portal, confirms that registration sub-district information is not available in the SNAC, nor are they included in the Index of Place Names, and digital boundary information is unavailable for sub-districts.
As a final resort I turned to the Code History Database (CHD). This database is easily accessible from the Open Geography Portal and viewed using Microsoft Access. Under the 'Geography History' section is a database of boundary changes that occurred after 2009. Registration sub-districts are included here, although by this time many sub-districts had been abolished. For the example given in the question, as the sub-district was still in existence in 2009, I found that code 6882 corresponds to Southwell sub-district, while 6881 was Newark sub-district. However, I was really lucky to find this information in the CHD as the vast majority of pre-2009 sub-districts are not included.
I contacted the GRO with this question, and this is their response:
I am afraid we are not able to forward you a listing of districts/sub
districts by name and number, such data is not available for sale from
the General Register Office.
You will find , however, that the Office for National Statistics
formerly the Office of Population Censuses and Surveys have in the
past produced publications entitled "Index of Place Names" for each
Census Year. These publications provide a district/sub district number
for each place name recorded. You should be able to purchase a copy
from The Stationery Office(TSO) formerly HMSO. Alternatively you may
be able to view some of these free of charge at The National Archives
Although GRO adopted the district /sub district numbers to form part
of the Index reference in 1993, the registration district numbering
arrangements go back many years to the very early census data. The
numbers will also have changed over the years but I can tell you that
688/1 and 688/2 were used for Newark area for the 1981 census up to at
A 2012 version of the Index of Place Names is available for free on the ONS website (linked above), however it does not include subdistrict information. Former versions may include this information, so it is an avenue to pursue.