Learn Russian: По подписании контракта мы сможем приступить к производству. - Upon signing the contract, we will be able to proceed to production.

While the preposition по usually takes the Dative case, it requires the Prepositional case when used to mean "upon" or "immediately after" an event. Because подписание is a neuter noun ending in -ие, its Prepositional ending is -ии.

Is по подписании common in everyday speech, or is there a more natural alternative?

This construction is highly formal and mostly used in business, legal, or official documents. In everyday spoken Russian, you would use после + Genitive instead: после подписания контракта (after the signing of the contract).

Why is контракта in the Genitive case instead of the Accusative?

Подписание is a noun, not a verb. While you would use the Accusative directly after the verb (подписать контракт), a noun modifying another noun requires the Genitive case. It literally translates to "the signing of the contract".

Why do we use the preposition к before производству?

The verb приступить (to proceed to, to get down to, or to start) strictly requires the preposition к followed by the Dative case. You will frequently see this verb in formal contexts when talking about starting tasks or phases.