We would all be having a very different conversation if MB were posting that the nanny was consistently 15-45 minutes late, and I doubt there would be any responses along the lines of "Well just think of her start time as 8:00 and then some days she will be super duper early!".
Actually changing the start time usually is a suggestion that gets posted several times when MBs post about nannies being late. Nannies also bristle at the notion of getting their pay docked for showing up late.
Its easier for MBs to find a new nanny than for a nanny to find a new job. Its even harder for a nanny without a current reference. OP is smart to think before acting.
I do agree that she should look but it will be harder without a reference. One point in her favor is that she has only been there since June. Its much more plausible that this isn't a fit issue. When nannies leave jobs that they have been at for a 1-3 years and can't produce a reference its a red flag that this is a mediocre nanny. NOt bad enough to fire but not great enough to give a reference. OP shouldn't stick it out for several years if this isn't a job that will yield a wonderful reference.