You are responsible for the well-being of your child, so if you don't feel comfortable leaving him with someone, trust your inner instincts. So what if your child can tell you if someone hurt them, the damage is already done. You may wish to get to know your neighbor better, invite them for dessert or coffee or dinner or brunch or something or even ask them ("things at work are changing and I might need to go in earlier, would you be willing...") They might say no and that's the end of it. You may discover a valuable resource, maybe do an exchange where you babysit for them occasionally in exchange.
As for work, who was suggesting you come in earlier, manager or coworkers? Is there a defined start time? Did you ever clear coming in later with your manager? If I had concerns, I'd setup a time to privately discuss it with my mananger to clarify expectations and what, if any flexibility there might be as far as your hours go. Not complaining, just approaching it from a "I've been a valuable employee here for 10 years, but there have been a lot of changes recently and I'm a conscientious employee, so I want to make sure that if my attendance is a problem, we can figure something out that will work for both of us" sort of way. You do not have to go into specifics, its none of their business. (You will have to be prepared for them saying that you must come in on time, and then you'll need to cross that bridge.) I do like the other person's suggestion of saying that coming in on time would require you to find alternate childcare with an insured provider for 15 minutes, and that it just might not be available (providers might want a minimum of a few hours). Co-workers can get jealous if they see someone getting special treatment (or maybe they have to do your job and theirs if you come in later which makes their job more stressful/difficult), so be mindful of that too. Good luck!