Finding the money supply graph for the UK
"What I would really like is the sum of instantly spendable British money that is currently possessed by all individuals and businesses"


The closest to what you want is $M1$ measure (see the informative table in the wikipedia article), but it seems that Bank of England does not provide it in its webpage (see the "explanatory notes" subpage if you want to learn what the different measures include).

This site provides the $M1$ measure and it appears you can export the data too.

On the same site, consider also the $M2$ measure. It includes "near-money" (i.e. assets that are quickly and easily converted into cash) and many economists consider it to be more representative of "money supply".

UPDATE
Here is the answer I received from Bank of England


  Our Bankstats tables can be found here -
  www.bankofengland.co.uk/statistics/pages/bankstats/current/default.aspx
  
  We publish M1 and M2 data in Bankstats table A2.3. Data is monthly and
  split into Seasonally adjusted and Not seasonally adjusted and then
  split again into Amounts outstanding, Changes and Growth rates. These
  are all shown on different tabs within the table.
  
  For more details about the components and construction of these money
  measures, please see the M3 explanatory note which can be found here
  www.bankofengland.co.uk/statistics/Pages/iadb/notesiadb/m3.aspx


The table they refer to has the title "$M3$" but if you download the spreadsheet you will see that it contains also $M1$ and $M2$ measures.