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but I'm not sure how this happens.
See my edits above. I guess Cyclescheme hires it from your employer (for free) so the extended use agreement is with Cyclescheme and it officially (temporarily) owns the bike. At the end of the extended use agreement you agree to buy the bike, Cyclescheme officially buys it from the employer (for the HMRC regulated price) and then you buy it from Cyclescheme for the same amount.
Greenbank
It makes it fairly clear in a number of places that Cyclescheme own the bike e.g.
but I'm not sure how this happens.