The 3D matter power spectrum, is a fundamental quantity in the analysis of cosmological data such as large-scale structure, 21 cm observations, and weak lensing. Existing computer models (Boltzmann codes) such as CLASS can provide it at the expense of immoderate computational cost. In this paper, we propose a fast Bayesian method to generate the 3D matter power spectrum, for a given set of wavenumbers, and redshifts, . Our code allows one to calculate the following quantities: the linear matter power spectrum at a given redshift (the default is set to 0); the non-linear 3D matter power spectrum with/without baryon feedback; the weak lensing power spectrum. The gradient of the 3D matter power spectrum with respect to the input cosmological parameters is also returned and this is useful for Hamiltonian Monte Carlo samplers. The derivatives are also useful for Fisher matrix calculations. In our application, the emulator is accurate when evaluated at a set of cosmological parameters, drawn from the prior, with the fractional uncertainty, centred on 0. It is also times faster compared to CLASS, hence making the emulator amenable to sampling cosmological and nuisance parameters in a Monte Carlo routine. In addition, once the 3D matter power spectrum is calculated, it can be used with a specific redshift distribution, to calculate the weak lensing and intrinsic alignment power spectra, which can then be used to derive constraints on cosmological parameters in a weak lensing data analysis problem. The software (emuPK) can be trained with any set of points and is distributed on Github, and comes with a pre-trained set of Gaussian Process (GP) models, based on 1000 Latin Hypercube (LH) samples, which follow roughly the current priors for current weak lensing analyses.