Is it faster to run straight to the windmill or to grab the speed buff first?

The path with the speed boost is quicker.

I tested this with my fiancee. We entered a custom game, chose the same character, and neither one of us purchased equipment. As such, our movement speeds were equal (320).

Then, we started at the very top corner of the stairs. This is the point your character naturally crosses when you go from the spawn to the top:

Green is actually slightly ahead since the two characters cannot occupy the same point. We put Green ahead to further illustrate the point.

At the same time, we ran to the top point. Green went around the speed boost (essentially not modifying the natural path that the champion takes when you click the top point), and red went through the speed boost (by way of manually guiding the champion across the speed boost plate). Red got their first with a little room to spare, even with the slight lead that Green had initially:

I just tested with 400 speed, going either way gave me 23 seconds, that's with no additional speed boosts. seems it's balanced in that way to not give an advantage.

without speed boost L: 9:51:50 A: 9:52:13

With speed boost L: 9:05:30 A: 9:05:53