Three points -
Bamber & June were 'barely' on speaking terms. The relationship breaking down years ago. June also disapproved of Julie's relationship with Jeremy. June had stopped financially supporting Bamber years ago. Would he ask to use her bike ? Would she say 'yes, or refuse outright, needing it for herself or just refuse because of the failed relationship ? Or both.
Is it realistic that Julie was supposed to make some of her way home on a bike ! Surely Bamber could fit in 15 minutes to drive her.
Is it a coincidence that the bike arrived just before the massacre.