From what I understand, BU moves the block size limit from consensus rules to a node policy rule. Instead of having the limit hard coded in, the user chooses their own block size limit. Also if a BU node detects a blockchain that has a higher block size (up to a certain user configurable threshold), after that chain is a number of blocks deep (user configurable), then it will switch to use that blockchain and set its block size limit higher.
So what happens if I left my node at 1MB +10% user threshold and a 1.2MB block comes - does my node reject it?
IIRC the node will keep the block and watch the chain it is on. If the chain it is on becomes n blocks deep (where n is user configurable) then your client will switch to use that chain as the active one. Otherwise it stays with the one it is currently using.
How will the network not split into a myriad little shards which diverge following accidental and/or intentional double-spends without manual human coordination?
I don't know. You'll have to ask someone else about that.