Cadenced optical imaging surveys in the next decade will be capable of detecting time-varying galaxy-scale strong gravitational lenses in large numbers, increasing the size of the statistically well-defined samples of multiply-imaged quasars by two orders of magnitude, and discovering the first strongly-lensed supernovae. We carry out a detailed calculation of the likely yields of several planned surveys, using realistic distributions for the lens and source properties and taking magnification bias and image configuration detectability into account. We find that upcoming wide-field synoptic surveys should detect several thousand lensed quasars. In particular, the LSST should find 8000 lensed quasars, 3000 of which will have well-measured time delays, and also ~130 lensed supernovae, which is compared with ~15 lensed supernovae predicted to be found by the JDEM. We predict the quad fraction to be ~15% for the lensed quasars and ~30% for the lensed supernovae. Generating a mock catalogue of around 1500 well-observed double-image lenses, we compute the available precision on the Hubble constant and the dark energy equation parameters for the time delay distance experiment (assuming priors from Planck): the predicted marginalised 68% confidence intervals are σ(w_0)=0.15, σ(w_a)=0.41, and σ(h)=0.017. While this is encouraging in the sense that these uncertainties are only 50% larger than those predicted for a space-based type-Ia supernova sample, we show how the dark energy figure of merit degrades with decreasing knowledge of the the lens mass distribution. (Abridged) Comment: 17 pages, 10 figures, 3 tables, accepted for publication in MNRAS; mock LSST lens catalogue may be available at http://kipac-prod.stanford.edu/collab/research/lensing/mocklens