The chemistry involved is initially the same as for orcein.   The same  precursors in the lichens are hydrolysed to orcinol which is oxidised by air in the presence of ammonia to give phenoxanone derivatives.   Under the more alkaline conditions, further oxidation and polymerisation occurs to give litmus.   The chemistry was finally understood after the work of Hans Musso as described on the  orcein page.

The 3-dimensional molecular structure of litmus is shown  on this page , but can only be viewed using Netscape.   Others might like to try  this picture.

Litmus has been made in The Netherlands at least since the 16th century. 

In 1940 this seeming monopoly came to an end when Johnsons of Hendon in the UK started producing it.
In the early days, Ochrolechia tartarea was the main lichen source, but later Rocella was used as well.  The process is very similar to that used for orchil except that potash, lime or gypsum were also added.
 
 

France
turnsole
tournesol
orseille en pierre
orseille de Flandres
orseille d'Hollande

Germany
persis 
lackmus

Scandanavia
lackmus

The Netherlands 
lakmoes

Details are difficult to find because the processes were kept secret.   This summary of a modern manufacturing procedure is from The vanishing lichens, D H S Richardson, London, 1975.

The lichens are ground in a solution of sodium carbonate and ammonia.   Stir the lichens from time to time and the colour changes from red to purple and finally blue after about four weeks.   The lichens are then dried and powdered.   At this stage the lichens contain partly litmus and partly orcein pigments.   The orcein is removed by extraction with alcohol, leaving the pure blue litmus.