Performing uncertainty quantification (UQ) for inverse problems requires repeated multi-scenario evaluations. When complex partial differential equations are involved, the high computational cost of full-order models (FOM), such as Finite Elements, makes traditional methods like Markov Chain Monte Carlo (MCMC) impractical. Additionally, relying on inexpensive data-driven surrogate models is challenging due to limited high-fidelity data and the high accuracy required for inverse problems. Conversely, low-fidelity data can be obtained more efficiently, simplifying regression tasks. Generally, a range of surrogate models can be applied to a given FOM problem. Deep Learning techniques allow for multi-fidelity fusion, combining information from various models to improve both efficiency and accuracy. Additionally, different models, each with unique performance characteristics, can be integrated into a UQ framework through a filtering multi-fidelity management approach. In this work, we propose a Multi-Fidelity Delayed Acceptance (MFDA) scheme that integrates these elements. Inspired by the Multi-Level Delayed Acceptance scheme, our method introduces a flexible framework combining solvers with different fidelities, not strictly related to simulation refinement levels. By leveraging progressive neural network surrogate models, this approach sequentially performs multi-fidelity regression from multiple sources, enabling on-the-fly selection of fidelity levels. In addition, it uses the FOM only during the training phase, allowing for coarser simulations at identification time. The strategy is tested on a set of benchmark problems, including isotropic groundwater flows and MEMS accelerometer calibration. Results show that our method consistently reduces the computational cost associated with MCMC sampling, offering a promising approach for efficient UQ in a wide range of engineering problems.