How would the next scheduled BP handle this mess?
There would just be blocks missing at those time points, and the next BP will build on top of what they know to be the block of highest height.
Relevant section from the whitepaper:
The EOS.IO software enables blocks to be produced exactly every 0.5 second and exactly one producer is authorized to produce a block at any given point in time. If the block is not produced at the scheduled time, then the block for that time slot is skipped. When one or more blocks are skipped, there is a 0.5 or more second gap in the blockchain.
If a producer misses a block and has not produced any block within the last 24 hours they are removed from consideration until they notify the blockchain of their intention to start producing blocks again. This ensures the network operates smoothly by minimizing the number of blocks missed by not scheduling producers who are proven to be unreliable.