Presumably if the hash algo is the same eg SHA256, then you can use that output where you want, and submit to either chain, while slotting in different blocks, BU and BTC around that result.
The hash
algorithm may be the same, but the
input to that algorithm is NOT the same.
When working on a BTC block, the input is the 80 byte BTC block header dependent on the list of BTC transactions in that BTC block.
When working on a BTU block, the input is the 80 byte BTU block header dependent on the list of BTU transactions in that BTU block.
Since most BTU transactions won't be valid in the BTC blockchain, and most BTC transactions won't be valid in the BTU blockchain, it wouldn't be possible to have both 80 byte headers be identical (since they are both dependent on different lists of transactions). Therefore, the output for a BTU block won't be valid for a BTC block (or vice versa).