Measurements of time delays between multiple quasar images produced by strong lensing are reaching a sensitivity that makes them a promising cosmological probe. Future surveys will provide significantly more measurements, reaching unprecedented depth in redshift, making strong lensing time delay (SLTD) observations competitive with other background probes. We forecast constraints on the nature of dark energy from upcoming SLTD surveys, simulating future catalogues with different numbers of lenses distributed up to redshift $zsim 1$ and focusing on cosmological parameters such as the Hubble constant $H_0$ and parametrisations of the dark energy equation of state. We also explore the impact of our ability to precisely model the lens mass profile and its environment, on the forecasted constraints. We find that in the most optimistic cases, SLTD will constrain $H_0$ at the level of $sim 0.1%$, while the CPL equation of state parameters, $w_0$ and $w_a$, can be determined with errors $sigma_{w_0}sim 0.05$ and $sigma_{w_a}sim 0.3$, respectively. Furthermore, we investigate the bias introduced when a wrong cosmological model is assumed for the analysis. We find that the value of $H_0$ could be biased up to $10 sigma$, assuming a perfect knowledge of the lens profile, when a $Lambda$CDM model is used to analyse data that really belong to a $w$CDM cosmology with $w=-0.9$. Based on these findings, we identify a consistency check of the assumed cosmological model in future SLTD surveys, by splitting the dataset in several redshift bins. Depending on the characteristics of the survey, this could provide a smoking gun for dark energy.
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