Overall, it looks good. Introducing the dedicated class and adding preprocessing transform feels a major improvement.
The following is my thoughts reading through the README and examples.

Major points

• Do we want to have parent the Weights class? I think the proposition can also be an idiom of best practice for the flexibility, which different project can implement without having a parent class.
  • check_type method is nice but it is simple enough (isinstance) to perform the check. Also what do you think of allowing the customize the behavior with duck-typing, say, in-house models?
  • The treatment of latest: bool feels a bit tedious for maintenance. When I add a latest model, I would like to be care-free about the previously-latest model. Since only one of them is supposed to be `True, how about simply making it a hard-coded class attribute?
• The idea of latest: bool might not synthesize well with audio. Say, I have a model architecture for Speech Recognition, I can have multiple of SOTA/latest models because of language / expected environments. (Note it is common to train and deploy multiple models with the same architecture, so that one is optimized for meeting room dictation, another is optimized for phone-conversation etc...)
• What if a model needs post-processing? Is the current framework extendable? For example, when the preprocessing is FFT, then post-processing can be the inverse of the specific FFT applied during the preprocessing.

Minor points

• The naming Weights feels slightly off because transforms not only contains weights but also defines the preprocessing operations as well.
• state_dict method should accept **kwargs, which will be passed to load_state_dict_from_url, so that the downloading process can be customized. (i.e. download location and such)
• I found it so hard to make Sphinx work well with Enum. Did you try?
• Currently, partial is used for the definition of transforms. I think this is good, because it will not instantiate an object. But every maintainer needs to be careful not to accidentally instantiate a transform here, and maintainers have to remember it at code review.