There was works between Gateshead Stadium and Pelaw recently notwithstanding the recent overhead wire incident there. Closing a major portion of the belt for a whole month would cause chaos but if it needs to be done, it will be. Haymarket is a possible for termination of service, recently there to FLE/Regent Centre they did works.
The best option really would be to shut the line from Heworth to Haymarket, allowing passengers with 3 zone tickets or DaySavers to transfer onto the next Northern service into Newcastle, at least letting them get to the next open portion of the line (albeit a considerable walk from Central).
There is a crossover between Monument and Haymarket which leads from the in line (platform 1) at Monument to the out line (platform 2) at Haymarket. Trains are turned at Haymarket should the line be closed north of Haymarket. So a train from South Shields would travel as normal to Haymarket, and would then wrong line out from Haymarket then across the crossover so when it gets to Monument it is on the normal running line. If the line is closed south of Monument a train from Airport would travel as normal to Monument, and then wrong line out of Monument and use the crossover so that when it reaches Haymarket it is back on the normal running line.
For the core section, trains from the north heading south towards Pelaw can be turned at South Gosforth, Jesmond, Monument, Gateshead Stadium, Heworth, and Pelaw. Trains from the south heading towards South Gosforth can be turned at Pelaw, Heworth, Haymarket and South Gosforth (they can then also be turned at Regent Centre/Regent Centre Sidings), you can also say Gateshead Stadium as trains would shunt forward empty towards Gateshead then turn around to use the crossover back into Stadium.
You can tell by those crossover points as to why disruption will be more severe to those who travel south of the river because there are considerably less crossover points.
I'm not sure if it's a full week/month long closure, but there is work booked in for that section of line. As of now, it just appears to be weekend closures but I'm not privy to the full engineering details.