Heimerdinger, Academy Instructor Carrion Dredger - [DEATHKNELL] Play a 1 Might Bird unit token with [DEFLECT] to your base. (When I die, get the effect. Opponents must pay 1 Rune to choose a [DEFLECT] unit with a spell or ability.) Does my mech token come in with Deflect? Sprite Fountain - [TEMPORARY] (Kill this at the start of its controller's Beginning Phase, before scoring.) When you play this, play a ready 3 Might Sprite unit token with [TEMPORARY] to your base. [DEATHKNELL] Repeat this gear's play effect. (When this dies, get the effect.) Does my mech token come in with Temporary?

No — neither keyword carries over: your Mech is a plain 3 Might Mech token, with no Deflect from Carrion Dredger's Bird and no Temporary from Sprite Fountain's Sprite.

Heimerdinger's effect replaces the whole token being played, so the replaced token's own characteristics (its Might, tags and keywords) never get applied — Deflect lives on the Bird token (Rule 187.7) and Temporary on the Sprite token (Rule 187.2), and neither belongs to a Mech token (187.4). What a replacement does inherit are modifications to the play event itself, not the token's identity (Rule 375) — which is why the 3 Might Mech token in the rules example keeps a "give it Temporary" grant from a linked ability, but here there's no such separate grant.

So from Sprite Fountain you do get one carry-over: "play a ready Sprite token" is an entry-state stipulation, so your Mech enters ready instead of exhausted (same for the Deathknell repeat). It just doesn't get killed at your next Beginning Phase.

As always with this Heimerdinger, the swap only works while he's at a battlefield — if he's at your Base you get the Bird/Sprite as printed.