Are Barry Allen and Bruce Wayne from different Universes?

You are making so many assumptions here. To make it clear there are two major universes.

The Arrowverse, which consists of the TV shows Arrow, The Flash, Legends of Tomorrow, Vixen and Supergirl (Supergirl exists in a different Earth ( Earth - 38) than Flash but is still canon to the Arrowverse). Constantine does exist on the same earth as Flash and Arrow.

But the movieverse is different and it's called DC Extended Universe, which consists of Man of Steel, Batman v Superman: Dawn of Justice, Suicide Squad, Wonder Woman for now and Justice League will come next. In DCEU there is no mention of any Supergirl or Arrow and there Barry Allen is a different interpretation of the same comic book character.

Here are official words on this approach of keeping TV shows and films separate from each other:

In October 2014, Warner Bros. announced the titles and release dates for nine films, along with some casting information. Also in the month, DC Comics' chief creative officer Geoff Johns explained DC's difference in approach to Marvel Studios and their cinematic universe, saying, "We look at it as the multiverse. We have our TV universe and our film universe, but they all co-exist. For us, creatively, it’s about allowing everyone to make the best possible product, to tell the best story, to do the best world. Everyone has a vision and you really want to let the visions shine through ... It's just a different approach. - Wikipeida

Even Supergirl season 2 is going to have their own version of Superman.

So there are not two conflicting timelines but two completely different universes or canons.

Refer here for the blog explaining the DC Cinematic Multiverse.

On the side-note Arrowverse did mentioned superman/batman and other DC superheros from Justice league time to time. Like Barry have Diana (Wonder Woman), Hal (Green Lantern) and Bruce on his speed dial on earth 2

And they did mentioned having Aquaman on Earth 2. And Earth one Legends of Tomorrow mentioned superman and batman in one dialogue

"I've seen men of steel die and dark knights fall.

But those are all Easter egg with no real appearance. But Legends of Tomorrow S02 is going to have there own version of of Justice Society of America

The interesting thing about season two is I think it's going to have a much, much different tone because our Legends are going to have a totally different purpose. They're actually going to have a totally different constitution. There will be new faces and new everything." The season will also introduce members of the Justice Society of America. The Society will consist of Vixen, Commander Steel, Obsidian, Stargirl and Dr. Mid-Nite. The season will also feature a version of the Legion of Doom, composed of Reverse Flash, Malcolm Merlyn, Damien Darhk, and Captain Cold - Wikipedia

But it have nothing to do with films, as they are separate entity for now. There is TV show Gotham too, centered around Gotham city but have nothing to do with any other TV show or Film yet and work individually.

It's subtly implied that they exist in some form, somewhere.

For example, in The Flash, Wells's "future newspaper" contained an article about a merger between Queen Consolidated and Wayne Industries. And in Legends of Tomorrow, Rip Hunter implies (without actually saying it) that the reason Superman and Batman aren't around is because Vandal Savage defeated (and presumably killed) both of them!

I have seen men of steel die and dark knights fall! Your entire 21st century world will suffer.