I have a nagging voice in my head saying that I shouldn't bother learning biochemistry, because it won't be useful in the long term because everything will be based on nanotech and we will all be uploads. Is that a valid point?

Keeping in mind the biases (EDIT: but also the expertise) that my username indicates, I would say that is nearly exactly backwards - modifications and engineering of biochemistry and biochemistry-type systems will actually occur (and already are) while what most people around here think of when they say 'nanotech' is a pipe dream. Biochemistry is the result of 4 gigayears of evolution showing the sorts of things that can actually be accomplished with atoms robustly rather than as expensive delicate one-off demonstrations and the most successful fine-scale engineering in the future will resemble it closely if not be it.