Green’s Dictionary of Slang

not care a tinker’s (curse) v.

also not give a tinker’s (curse), not care/give a tinker’s blast, ...continental, ...cuss, ...damn, ...darn, ...fart, ...a tinsmith’s malediction
[the lack of importance one gives a curse thrown over the shoulder of a departing tinker who has been unable to sell anything or find work; note letter 08/02/1999 from Mr George Shaw: ‘I learnt my trade as a Tinker and Plumber, and I used a tinkers cuss many, many times. When the copper kettle was heated on the open fire, the fire would burn a hole in the bend at the bottom of the spout. The tinker would take a piece of bread and ram it into the spout blocking up the hole. He would then proceed to solder a patch onto the hole . The bread would dam up the hole, hence a tinker’s cuss. This was also a way the plumber would use to soak up the water in the lead pipe (if there was a small seepage of water) whilst he wiped a joint. On both occasions, once the jobs were finished, the water would wash out the bread; the OED, however, dismisses a similar ety. for tinker’s dam(n) as ‘ingenious but baseless’]

[mid-18C+] to not care at all.