We present the first Love-wave group velocity and shear velocity maps of the British Isles obtained from ambient noise interferometry and fully non-linear inversion. We computed interferometric inter-station Green's functions by cross-correlating the transverse component of ambient noise records retrieved by 61 seismic stations across the UK and Ireland. Group velocity measurements along each possible inter-station path were obtained using frequency-time analysis and converted into a series of inter-station traveltime datasets between 4 and 15 seconds period. Traveltime uncertainties estimated from the standard deviation of dispersion curves constructed by stacking randomly-selected subsets of daily cross-correlations, were observed to be too low to allow reasonable data fits to be obtained during tomography. Data uncertainties were therefore estimated again during the inversion as distance-dependent functionals. We produced Love-wave group velocity maps within 8 different period bands using a fully non-linear tomography method which combines the transdimensional reversible-jump Markov chain Monte Carlo (rj-McMC) algorithm with an eikonal raytracer. By modelling exact raypaths at each step of the Markov chain we ensured that the non-linear character of the inverse problem was fully and correctly accounted for. Between 4 and 10 seconds period, the group velocity maps show remarkable agreement with the known geology of the British Isles and correctly identify a number of low-velocity sedimentary basins and high-velocity features. Longer period maps, in which most sedimentary basins are not visible, are instead mainly representative of basement rocks. In a second stage of our study we used the results of tomography to produce a series of Love-wave group velocity dispersion curves across a grid of geographical points focussed around the East Irish Sea sedimentary basin. We then independently inverted each curve using a similar rj-McMC algorithm to obtain a series of one-dimensional shear velocity profiles. By merging all 1D profiles, we created a fully three-dimensional model of the crust beneath the East Irish Sea. The depth to basement in this model compares well with that averaged from seismic reflection profiles. This result is the first 3-dimensional model in the UK with fully quantified uncertainties: it shows basin depths and basement structures, and their concomitant uncertainties.