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We present the CO isotopologue Line Atlas within the Whirpool galaxy Survey (CLAWS) based on an IRAM 30-m large programme which provides a benchmark study of numerous, faint CO isotopologues in the mm-wavelength regime across the full disc of M51 (NGC 5194). The survey's core goal is to use the low-J CO isotopologue lines to constrain CO excitation and chemistry, and therefrom the local physical conditions of the gas. In this survey paper, we describe the CLAWS observing and data reduction strategies. We map the J=1-0 and 2-1 transitions of the CO isotopologues $^{12}$CO,$^{13}$CO, C$^{18}$O and C$^{17}$O, as well as several supplementary lines within the 1 mm and 3 mm window (CN(1-0), CS(2-1), CH$_3$OH(2-1), N$_2$H$^+$(1-0), HC$_3$N(10-9)) at ~1 kpc resolution. A total observation time of 149 h offers unprecedented sensitivity. We use these data to explore several CO isotopologue line ratios in detail, study their radial (and azimuthal) trends and investigate whether changes in line ratios stem from changes in ISM properties such as gas temperatures, densities or chemical abundances. For example, we find negative radial trends for the $^{13}$CO}/$^{12}$CO, C$^{18}$O/$^{12}$CO and C$^{18}$O/$^{13}$CO line ratios in their J=1-0 transitions. We also find variations with local environment, such as higher $^{12}$CO(2-1)/(1-0) or $^{13}$CO/$^{12}$CO(1-0) line ratios in interarm regions compared to spiral arm regions. We propose that these aforementioned variations of CO line ratios are most likely due to a variation of the optical depth, while abundance variations due to selective nucleosynthesis on a galaxy-wide scale could also play a role. We also study the CO spectral line energy distribution (SLED) using archival JCMT $^{12}$CO(3-2) data and find a variation of the SLED shape with local environmental parameters further underlying changes in optical depth, gas temperatures or densities.