The Genesis block, handcrafted with crypto passion by Satoshi Nakamoto, is block height 0. Computer nerds always start counting from zero.
And _act_ was quicker: first block with subsidy of 3.125
BTC is block height 840,000 and this block subsidy will last in the 5
th epoch until block height 1,049,999 (inclusive).
Does it begin when miners start working on block 840000 or when it is found?
Your question doesn't make sense to me. A block of height 840,000 is only accepted and valid when its coinbase transactions has a subsidy of max. 3.125
BTC plus transaction fees. To crunch hashes on a valid block candidate, miners have to assemble a correct coinbase transaction and all the rest of the block before their ASICs start hashing, otherwise it's a waste of time, money, energy, sense.
Edit (as franky1 points out correctly):
Miners who mine in pools rely on the pool operator to assemble a proper block candidate and double hash variations of the 80-bytes block header.