I'm a welder. I know the answer to this question is that there are many older businesses than welding. I'm not sure I buy the thesis that prostitution is older because I know of no records for sex for money that antedate the sale of food or furniture or metals for which we have receipts going back more than 5000 years.
I bring up the issue for two reasons. Welding is by definition the melding of two metals by liquefying them both and puddling the liquid pool. This was done by blacksmiths as far back as three millennia. Japanese sword makers were able to make swords by welding brittle metal for the edge with more flexible metal for the core. I saw welding being done in Kabul in 1971 where the acetylene was in a clump of rock and the fumes were picked up in a hood and connected with a hose to the torch where it was merged with oxygen.
I would be happy if someone looked into this issue and published the results.
What leads me to think about it is that most industries do not have a very long lifespan. Metallurgy may be one of the exceptions that survives into the modern world and has been able to adapt because it is not associated with any single company nor trade and it must adapt to almost every other technology in order to survive: bridge building, engine building and transportation.