The Welsh All Blacks are determined to bounce back after failing to win the Principality Premiership for the first time since the league turned semi-professional.
Alongside Brew and Bishop, Neath also signed former Newport Gwent Dragons and Carmarthen Quins winger Jamie Davies.
"We are very pleased with the capture of three very talented three-quarters", said Neath Head Coach Patrick Horgan.
"They will add top quality and a cutting edge to the good work of our pack. I am looking forward to working with them.
"Last season, our tries per game average fell just below four for the first time in the semi-pro era and, while defences get ever tighter and sides become ever more determined to stop us playing, we feel that these three signings will ensure that we play the kind of all-out attacking rugby that is our hallmark at The Gnoll.
"Any further behind the scrum signings will be limited to opportunities for local youngsters.
"We have a few training with us regularly already and I would invite any other promising lads who feel they can make the grade to get in touch."
As Brew, Bishop and Davies arrive at The Gnoll, the quartet of Steve Thomas, Greg Evans, Gerwyn Price and Ejike Uzoigwe depart.
"It is a fact of rugby life that as new players arrive, others must depart," added Horgan.
"We sincerely thank the departing players for their efforts for Neath and we wish them every success in the future."
Ex-Ospreys centre, Bishop, appeared 22 times for the Swansea-based region, after joining in 2004, scoring four tries.
The 27 year-old was close to making the Wales squad under Mike Ruddock.
But the Treorchy product suffered a shattered knee-cap against Munster to dash his international hopes. Bishop later joined the Sale Sharks in 2009 and has since enjoyed brief spells at the Dragons and Edinburgh.
Nathan Brew also returns for his second spell at Neath having won the Premiership with the All Blacks during the 2009/10 season.
Brew started his career at Newport RFC in 2001 before making the move to Newport Gwent Dragons in 2003 with the introduction of regional rugby.
He became a crowd favourite at Rodney Parade playing mainly on the wing before being moved to centre in 2006/07 season.
Brew's single Welsh cap came in 2003 when he appeared in the 54-8 victory over Romania in Wrexham, claiming a try in the process.
Brew moved to the Scarlets in 2007, but found first-team opportunities limited. He has since played for Bristol and enjoyed a second stint at the Dragons.