You're probably right about the day of transfer, though a phonecall to them wouldn't have gone amiss, just to get it sorted out. I recall when changing my car (at about 11am/mid-day) that my insurer could change them time when it transferred over, but had to be done over the phone.

If you're not driving your old car away, then you should have no problems if the new policy starts at midnight on Sunday - I had to drive up to Motorpoint's Derby premises from Hertfordshire to PX my old car and pickup the new one, so I waited until noon before driving away.

When it comes to insurance, always check with you're insurer over the phone - don't assume (especially using website - they're notoriously ambiguous about such things), given that if the police pull you over and you're not insured (even if via an oversight), you'll pick up at least 6 points and no excuses will do (as if you'd had an accident, you wouldn't be covered, nor would [if your fault] the no-fault other driver, who would have to pay for damage to their car.