The KEX schemes that seem to have received the most cryptanalysis are SIKE (SIDH that permits key reuse) and NTRU. They seem solid but I’d only use them in the real world in a hybrid scheme where the key is hashed with the result of a conventional ECC exchange. That way you get that security if the PQ algorithm ends up broken.

The signature schemes seem dodgy to me except for Sphincs and it’s variants and those have big keys and signatures. The keys are not impractically big for many uses but would be tough for things like block chains.