Learn Polish: Wczoraj spuchł mi nadgarstek. - Yesterday my wrist swelled up.

This is the dative of inalienable possession. In Polish, when an action affects your own body parts, health, or close family, you use a dative pronoun (like mi) to show who experiences the effect, rather than a possessive word (like mój) to show ownership. Using mój here would sound unnatural, as if the wrist were a detached object you own.

What case is nadgarstek in this sentence?

It is in the nominative case. Grammatically, nadgarstek (the wrist) is the subject doing the action of swelling up (spuchł). The English translation says 'my wrist swelled up,' but the Polish structure is literally 'the wrist swelled up to me.'

What is the infinitive of spuchł, and why isn't it 'spuchnął'?

The infinitive is spuchnąć (to swell up). Many perfective verbs ending in -nąć that describe a physical change drop the -ną- part in the past tense. Therefore, the third-person singular masculine form is spuchł rather than the regular-looking but incorrect 'spuchnął'.

Could I put mi at the beginning of the sentence, before wczoraj?