Applications of Takens' theorem to TCS?
Takens himself did some CS work although not TCS work. He did some attractor reconstruction stuff with neural networks, for example (https://clgiles.ist.psu.edu/papers/NC-2000-learning-chaos-nn.pdf) and it does use his embedding theorem to suggest nonlinear autoregressive (time delay) neural nets, and look towards actual recurrent networks.

Overall, dynamical systems is closest to CS in neural network land because of the easy dynamical interpretation of the models. Try some of the Ganguli gang's publications (https://ganguli-gang.stanford.edu/pubs.html), esp. Exponential expressivity in deep neural networks through transient chaos


  apparently there have been recently methods designed to find these in a more automated way


Again, not theoretical, but I've used them for non-example time series data and they aren't incredible in practice, by the way. The denoising aspects of them is actually quite difficult.