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Neutron spectroscopy is an invaluable tool for many scientific and industrial applications, including underground Dark Matter searches. Neutron-induced backgrounds produced by cosmic ray muons and the cavern radioactivity can mimic the expected Dark Matter signal. However, existing neutron detection methods have several drawbacks and limitations, thus measurements remain elusive. A promising new approach to neutron spectroscopy is the use of a nitrogen-filled spherical proportional counter that exploits the $^{14}$N(n,$α$)$^{11}$B and $^{14}$N(n, p)$^{14}$C reactions. This is a safe, inexpensive, effective and reliable technique. In this work, the latest instrumentation developments are incorporated in a compact detector operated at the University of Birmingham (UoB) with high gain at gas pressure up to 1.8\,bar. We demonstrate spectroscopic measurements of thermalised and fast neutrons respectively from an $^{241}$Am-$^9$Be source and from the MC40 cyclotron facility at UoB. Additionally, the detector response to neutrons is simulated using a framework developed at UoB and compared with the experimental results.