There's surprisingly little in the way of serious scholarship on this, certainly not that I've been able to find. There are a number of articles and think-pieces online, though, and they agree on the basics.

• The first ever club to have a shirt sponsor was Peñarol in Uruguay in the 1950s.

• In the 1960s, sponsorship was permitted by the Danish and Austrian football associations, and in some French leagues.

• The first major sponsorship deal was in 1973-74 with Eintracht Braunschweig in the German Bundesliga, who got around the ban by adopting a new club badge that just happened to be the Jaegermeister logo and putting a large "club badge" on the front of their shirts. Bayern Munich pulled something similar by replacing their club badge with the name of the kit manufacturer and having that front, centre and large. The German FA gave up and legalised sponsorship.

• During the late 70s and early 80s most European leagues caved to the pressure and allowed sponsorship, though there were some holdouts. Italian clubs tended to be more traditionalist and not all were sponsored until the mid-90s, while Barcelona remained a last bastion of proud non-sponsorship until 2006.

There's a rather good article here that goes into a lot more detail.