Industrial IoT (IIoT) is one of the major verticals targeted by the next generations of wireless networks. In order to provide industrial plants with features relying on wireless communications, the grant-free RA (GFRA) protocol appears to be a promising means for supporting massive ultra-reliable connectivity; at the same time, it is a critical bottleneck that requires an access point (AP) to be able to jointly perform active user detection and channel estimation (AUDaCE) to fulfill its main mission of allowing industrial wireless devices to access the core network. This mission is even harder when the GFRA requests are correlated because of event-driven activity triggers. This paper proposes a new tractable gaussian correlated activity (GCA) model for this scenario. The corresponding AUDaCE problem is then studied in the Bayesian compressed sensing (BCS) framework. An hybrid instance of the generalized AMP (GAMP) algorithm is derived and its capability to perform AUDaCE is numerically assessed by extensive Monte-Carlo simulations. The numerical results show gains of 2.5dB in channel estimation gain for twice less detection errors w.r.t. state-of-theart algorithms.