The simulation of complex multi-physics phenomena often relies on System of Solvers (SoS), which we define here as a set of interdependent solvers where the output of an upstream solver is the input of downstream solvers. Performing Uncertainty Quantification (UQ) analyses in SoS is challenging as they generally feature a large number of uncertain input parameters so that classical UQ methods, such as spectral expansions or Gaussian process models, are affected by the curse of dimensionality. In this work, we develop an original mathematical framework, based on Gaussian Process (GP) models, to construct a global surrogate model of the uncertain directed SoS, (i.e. merely featuring one-way dependences between solvers). The key ideas of the proposed approach are i) to determine a local GP model for each solver constituting the SoS and, ii) to define the prediction as the composition of the individual GP models constituting a system of GP models (SoGP). We further propose different adaptive sampling strategies for the construction of the SoGP. These strategies are based on the decomposition of the SoGP prediction variance into individual contributions of the constitutive GP models and on extensions of the Maximum Mean Square Predictive Error criterion to system of GP models. The performance of the SoGP framework is then assessed on several SoS involving different numbers of solvers and structures of input dependencies. The results show that the SoGP framework is very flexible and can handle different types of SoS, with a significantly reduced construction cost (measured by the number of training samples) compared to the direct GP model approximation of the SoS.