Should the hay field that has not been cut the second time in the same year be brush hog before winter

That would seem to me to be a "Depends" type answer. If you do cut it then the spring will bring new growth with no tall stalks to overcome to get sunlight. If you don't cut it there will still be new spring growth but it will have to reach for the sun through the winter knocked down, long blades.
There is a third thing in this case and that is that there will be more above ground level rhizomes if it isn't cut before winter. I think either way works, just depends on if you want more grass nodes in the soil for next year (thicker growth).

Bermuda is a funny grass plant, if you let it grow with no cutting it will spread by seed heads, but if you cut it rather short it will multiply like wildfire and become a thick mat of grass plants.

Garry Rausch wrote:Should the hay field that has not been cut the second time in the same year be brush hog before winter

It really depends where a person lives. If they are in northern lattitudes like where I live here in Maine, the biggest thing to watch out for is that there is suffecient height of the sward to survive through winter. For instance I never plant grass after August 1st otherwise its roots would not be established enough to survive the winter.

I have cut fields in September for my second crop with no problem, but cutting it in October stunted its growth the next year!

Timothy is a strange type of grass though. While Cut and Drop is a great method, it does not work so well for Timothy, but if your field is teeming with 100% Timothy, there is no real reason to use Cut and Drop either; you already have a nice field.

Cutting will speed up growth in the spring but if you drive the tractor around while the soils wet you will compact it slowing it down. Also why dont you try to get some legumes in that field.