Are there quantum algorithms demonstrating speedup computing classical neural networks (in 2021)?

It seems clear that any speedup for neural networks will be inherently polynomial in the traditional sense – they don't traditionally use Fourier transforms or any other subroutines that we often associate with exponential speedups. But if we look to things related to training time (e.g. sample complexity) and learnability, we may find useful advantages in practice. I shared a link at the end of my answer to a related question that may be of interest.

Of course in terms of implementation and hardware, classical deep learning is getting pretty advanced. It may take a long while for any quantum advantages to be realized in practice.