  This work attains a threefold objective: first, we derived the richness-mass
scaling in the local Universe from data of 53 clusters with individual
measurements of mass. We found a 0.46+-0.12 slope and a 0.25+-0.03 dex scatter
measuring richness with a previously developed method. Second, we showed on a
real sample of 250 0.06<z<0.9 clusters, most of which are at z<0.3, with
spectroscopic redshift that the colour of the red sequence allows us to measure
the clusters' redshift to better than Delta z=0.02. Third, we computed the
predicted prior of the richness-mass scaling to forecast the capabilities of
future wide-field-area surveys of galaxy clusters to constrain cosmological
parameters. We computed the uncertainty and the covariance matrix of the
(evolving) richness-mass scaling of a PanStarrs 1+Euclid-like survey accounting
for a large suite of sources of errors. We find that the richness-mass scaling
parameters, which are the input ingredients of cosmological forecasts using
cluster counts, can be determined 10^5 times better than estimated in previous
works that did not use weak-lensing mass estimates. The better knowledge of the
scaling parameters likely has a strong impact on the relative importance of the
different probes used to constrain cosmological parameters. Richness-mass
scaling parameters were recovered, but only if the cluster mass function and
the weak-lensing redshift-dependent selection function were accounted for in
the fitting of the mass-richness scaling. This emphasizes the limitations of
often adopted simplifying assumptions, such as having a mass-complete
redshift-independent sample. The fitting code used for computing the predicted
prior, including the treatment of the mass function and of the weak-lensing
selection function, is provided in the appendix. [Abridged]
