I'm assuming that they haven't had contact, but is there any information on the relationship with the East Coast Brotherhood of Steel and the NCR?

No none, actually with the Brotherhood war going on at the time that Lyons crew went east the NCR probably doesn't even know they sent a expedition in the first place.

Lyons would know of NCR but I doubt NCR would know of Lyons.

Still if Lyons was sent to DC to research new super mutants in DC, NCR would have heard the same rumors the BoS did.

Highly doubt it. Two different coasts after all.

"After the defeat of the Master, the original Brotherhood found themselves at odds with their need for new blood versus their code of technological secrecy. The debate went lightly. Finally, the elders ruled against sharing the technology with outsiders, convinced that they would endure what they had before. Further discussion was discouraged when the elders ordered the minority who wished to share their technology on a mission across the wastes. The Brotherhood constructed airships and dispatched the minority East, to track down and assess the extent of the remaining super mutant threat."
Copypasta from the wiki.