This is the first paper in a series that present a multi-wavelength analysis of the archetype Ultra-Luminous InfraRed Galaxy (ULIRG) IRAS FSC10214+4724, a gravitationally lensed, starburst/AGN at z=2.3. Here we present a new lens model and spatially-resolved radio data, as well as a deep HST F160W map. The lens modelling employs a Bayesian Markov Chain Monte Carlo algorithm with extended-source, forward ray-tracing. Using these high resolution HST, MERLIN and VLA maps, the algorithm allows us to constrain the level of distortion to the continuum spectral energy distribution resulting from emission components with differing magnification factors, due to their size and proximity to the caustic. Our lens model finds the narrow line region (NLR), and by proxy the active nucleus, is preferentially magnified. This supports previous claims that preferential magnification could mask the expected polycyclic aromatic hydrocarbon spectral features in the Spitzer mid-infrared spectrum which roughly trace the star-forming regions. Furthermore, we show the arc-to-counter-image flux ratio is not a good estimate of the magnification in this system, despite its common use in the IRAS FSC10214+4724 literature. Our lens modelling suggests magnifications of μ ~ 15-20+-2 for the HST F814W, MERLIN 1.7 GHz and VLA 8 GHz maps, significantly lower than the canonical values of μ = 50-100 often used for this system. Systematic errors such as the dark matter density slope and co-location of stellar and dark matter centroids dominate the uncertainties in the lens model at the 40 percent level. ; Comment: Dedicated to the memory of Steve Rawlings. Accepted for publication in MNRAS. 20 pages, 15 figures