How can Wells know what happened in the alternate time?

Wells has had cameras in every location of note. He was monitoring what the scoobies were planning. And Cisco's very vivid lucid dreaming included all the details he needed to know in order to have Everyman recreate the scene close enough to scare Cisco who only half remembered it.

Additionally, in both comics and the show, Eobard, like most speed force speedster are somewhat immune to time changes, similar to how Cisco was portrayed. Flashpoint Paradox is a prime example. Zoom who was not involved in Barry's saving his mom, had no memory loss from the alternative time line.

finally, Gideon, the advanced A.I. is able to track changes in the timeline. This can help Wells fill in any holes from what he learned from Cisco's recorded dream.

As cde mentions in his answer, Wells did have cameras all over the place (the Cortex, Barry's lab, Iris' desk at work, the West/Allen house and numerous other places), but that doesn't fully answer your question, as he obviously couldn't see events on there that never happened (or rather that were changed).

cde also correctly mentions that the Speed Force protects speedsters from changes in the timeline - to a degree at least - but as you point out in your question, Eobard was genuinely surprised when Cisco reported he (ET) killed him (CR).

So, while Eobard had Barry's lab bugged (which is where they brought the body of the real Harrison Wells and did Cisco's trip down alternate memory lane, after all), he knew they were onto him.

Now it gets a bit complicated (which means I have to go a bit of a long way to get to my point):

Eobard was aware of that much earlier then strictly makes sense (IMO), because Joe told Cisco about his suspicion about Wells in Barry's lab, yet Eobard did nothing to prevent them finding out anything more, probably because he thought it would accomplish two things:

• divert suspicion from him, since only Eobard's DNA and not Well's could be found and
• it would get the idea of time travel into Barry, pushing him to go faster, with the possible goal in mind of saving his mom, and Eobard needs Barry to reach that speed in order to get back home.

Barry told Joe he believed Joe's theory, also in the lab, and when they decided to go to Starling City they did so in the West/Allen living room, yet here again Eobard did nothing to move Well's body or mask his tracks. He did however move up his timetable a bit, since everything (Grodd, Everyman, the rebuild particle accelerator) was falling into place and he probably believed Barry being able to be fast enough by then.

Having done this rather extensive detour I can finally coming round to answering what you're specifically asking, just after mentioning the rather important fact that they also did Cisco's "dream walk" in Barry's lab, so the important points (vibrating hand through heart, calling Cisco a son, confessing Nora's murder, but stating that that had not been the intention)

How then did Everyman know what to say? How could it match almost verbatim with what was said in the alternate timeline?

Eobard told him what to say, simple as that. Like the hologram of Reverse Flash, Every"wells" said, what Eobard would have said, had he actually been there. The alternate timeline Eobard did speak from the heart and told Cisco what he truely felt, which is after all what the real Eobard felt (changing one day didn't alter that significantly enough), so he was able to give Everyman an accurate enough "script" to follow ("Nora wasn't the intended target", "Cisco is like a son", walking up menacing with hand outstretched).