Although difficult to discuss not knowing the facts of the proposed TL, how long would it take to be able to construct fibreglass body armour in large enough quantities to partially equip several infantry battalions? Would a two to three year TL seem appropriate or would it be earlier or later?
Well now that's an interesting question...
Let's take 1893 as our starting point; the year in which one Edward Drummond Libbey (a glassmaker from Toledo
[1]) exhibited a dress incorporating glass fibres with a diameter and texture approximating that of silk fibres at the Chicago Columbian Exposition. Say someone notices, talks to Libbey and by 1895 the process that Slayter discovered has been developed. I don't believe it would be impossible with the technology of the period.
Now, as Carl Schwamberger pointed out, there were earlier alternatives to polyester (I studied chemistry so I tend to think first of synthetics, the idea of using polymerised natural oils just didn't occur to me) that could form suitable resins. It's been a couple of decades since my materials science classes but I believe that a couple of years of experimentation could yield a useful material incorporating glass fibres in a resin. Say by 1902
[2] the material is in limited production and people are wondering what to do with it.
Then, the following year a couple of bicycle makers developed a contraption that allowed them to fly, in a somewhat controlled manner, for a short distance through the air.
[3]
So let's assume the big bits of history plot along more-or-less as OTL and the Great War starts on schedule in 1914
[4]. At this stage I'd say the new fibreglass would be in production on a reasonable scale, for things like aircraft elements and motor torpedo boats, and could start to be used for splinter vests, at least on a small scale initially.
BTW if anyone does develop a timeline based on this idea, don't forget to include Amos Root. Another Ohioan and a fascinating character.
[1] Interestingly Libbey employed one Michael Owens and helped found the precursor companies to Owens-Illinois where Slayter would develop fibreglass OTL.
[2] Which by an interesting coincidence is the year in which the city of Toledo failed to host the Ohio Centennial and Northwest Territory Exposition. Maybe in this slightly divergent timeline that World's Fair goes ahead. Perhaps this button is made of the new wonder material?
[3] Interesting how many connections the brothers have to Ohio isn't it? Hmmm.
[4] Though maybe FF survives thanks to an experimental bullet resistant waistcoat?