A new framework for spatio-temporally adaptive turbulence simulations is proposed. The method is based on a variable-fidelity representation that tightly integrates numerics and modelling of subgrid-scale turbulence and aims to capture the flow physics on a near-optimal adaptive mesh. The integration is achieved by combining hierarchical wavelet-based computational modelling with spatially and temporally varying wavelet threshold filtering. The proposed approach provides automatic smooth transition from directly resolving all flow physics to capturing only the energetic/coherent structures, which leads to a dynamically adaptive variable-fidelity approach. The self-regulating continuous switch between different fidelity regimes is accomplished through a two-way feedback mechanism between the modelled dissipation and the local grid resolution, which is based on spatio-temporal variation of the wavelet filtering threshold. The proposed methodology systematically accounts for and exploits the spatial and temporal intermittency of turbulence. Thus, it overcomes the major limitation of all existing wavelet-based multi-resolution techniques, namely, the use of a global thresholding criterion. The procedure consists of tracking the wavelet thresholding factor within a Lagrangian frame by exploiting a Lagrangian path-line diffusive averaging approach, based on either interpolation along characteristics or direct solution of the corresponding evolution equation. This new methodology is tested for linearly forced homogeneous turbulence at different Reynolds numbers and provides very promising results on a benchmark with time-varying prescribed level of turbulence resolution.