It depends on where you put the end points, overall you could easily make the claim that somewhere in the ballpark of 25 to 50 years is as good a figure as any.

Also, it's worth noting here that the horse to car transition wasn't the only one. You also have the introduction and adoption of bicycles (which drove the creation of higher quality hard surfaced roads before cars did) as well as long-range and local rail (e.g. subways, streetcars, and elevated trains), all of which happens in a similar timeframe. The first automobiles came about starting in the 1870s, which was coincidentally also around the same time as the first elevated trains and subways (1860s) as well as the first widely popular bicycles (penny-farthings in the 1870s then the safety bicycle in the 1880s and later).

Regardless, it took a long time for automobiles to become ubiquitous. It took until the late 1880s for the motorcar to transition from curiosity and toy to a commonplace and practical mode of transportation for goods or people. Though proper gas stations didn't exist until just after the turn of the century (prior to which gasoline and diesel would have been purchased in containers from pharmacies). Even during WWI (and to a lesser extent WWII) horses were commonplace components of civic and military infrastructure and logistics. But, by the 1910s it was pretty clear that horses were becoming obsolete. In New York City specifically it was in the 1910s where cars began to outnumber horses and also in that decade when the last horsecar (a streetcar on rails pulled by horses) was decommissioned.

So, depending on how you measure you have perhaps 25 years from the late 1880s to the mid 1910s where cars went from curiosity to part of the urban infrastructure or nearly 50 years starting from the early 1870s to the late 1910s where cars transitioned from their earliest forms to dominant forms of transportation.