Our work proposes a characterisation and testing methodology, as well as the relevant custom implementation, for measuring novel digital security circuits that use nanomaterial-based Physical Unclonable Functions (PUFs) as their security anchors. Although in this work we focus on PUFs that utilise the electrical characteristics of a crossbar structure of Carbon NanoTube (CNT) cells, the proposed methodology is applicable to most, if not all, PUFs that are based on similar crossbar structures of nanomaterials. Our work describes and discusses in detail the relevant characterisation and testing framework, while also presenting the corresponding mixed-signal circuit implementation, which can be utilised to provide a digital security token in an automated manner. Finally, preliminary results concerning the considered CNT PUFs are also presented, proving in this way the ability of the proposed framework to be utilised for the characterisation and testing of these PUFs, as well as for the implementation of security applications in the context of embedded systems and the Internet of Things (IoT), using nanomaterial-based PUFs in general.