The very short answer to this is no. There are many reasons why there weren't any though.
There are many stories of tunnels running from Peterborough Abbey in the city's folklore. It seems any significant building that is within a few miles of the abbey/cathedral has been rumoured to be linked to the abbey by a tunnel. Oxney House, supposedly had a tunnel, Thorpe Hall and the associated holy well was strongly rumoured to have a tunnel, and another was said to run from the cloister well to the river. There are even rumours of tunnels running all the way to Crowland and Ely Abbeys - the mind boggles!
There are tunnel-like structures underneath the cathedral, which relate to earlier buildings and repairs carried out. Accounts of people crawling through tunnels are true, but these tunnels don't continue outside of the building. Any tunnels stretching from under the abbey would have to be built at the same time as it was, so as not to undermine it, making them 900 years old. This in itself sounds problematic! The tunnels would have to be made from stone if they were to last any length of time, which would have been a significant waste of manpower when the abbey was being built. If made of wood, they would have collapsed almost instantly given the abbey's proximity to the Nene and raised water table - soggy wood doesn't make for good tunnels!
The water table (where the underground water level meets the soil) in Peterborough is exceedingly high in places and many locations used to sit underwater in the winter, such as the land between the abbey and the river. One of the reasons that St John's church was moved to the market place in the 15th century was because the ground around it was waterlogged in winter. Add in the fact that the river used to be tidal and the fens were constantly soggy except on islands such as Whittlesey and Thorney, you would likely have needed scuba gear to use any tunnels that headed south or east from the abbey. In addition, the market place (now Cathedral Square) was often sodden until the ground level was significantly raised, suggesting that even a tunnel heading west was liable to flooding.
If a tunnel running west from the abbey was feasible, what would be the logistics of creating a tunnel from the abbey to Thorpe Hall? Firstly, they would have needed enough stone for a tunnel one-and-a-half miles long and the manpower to both dig the trench and line it with stone. But how big and how deep would the tunnel need to be? If it's for monks to flee, it would probably want to be a full height tunnel, but even at a crawling height, they would have to dig down several feet to place the tunnel and backfill the ground above so it wasn't visible. In a city centre like Peterborough, there is no location that a tunnel could have been placed that it would not have been discovered, for foundations for modern buildings, and utility trenches, go down a long way and would have disturbed it. This is before we factor in the railway and the building of the Thorpe Road subway in the 1880s - if there was a tunnel, they would have recorded this fact.
Even if a tunnel stretching one-and-a-half miles, dug several feet into the ground 900 years ago, could theoretically survive, the other issue is the geology of the area. The land between the abbey/cathedral and Thorpe Hall is heavy with thick clays, cornbrash, and limestone deposits; anyone who has attempted to dig those in their garden knows they are not easy to dig. In addition, the ground is not flat but varies in height causing another issue. If you've ever walked the land around Thorpe Hall, there's a formidable hill to tackle if you're digging a tunnel.
So unfortunately, all of the tales of tunnels and fleeing monks are nothing but stories, but who doesn't love a good story?!
Image by Peter H from Pixabay