It's important that your doctor wash their hands, but less important than you might think. As a health care provider, I wash my hands before examining every patient, but it's out of respect for the patient's feelings, not because my hands are particularly dirty (I wash my hands after every patient encounter. I'm unlikely to bring you that patient's germs.)
Trust me, the germs are everywhere, not only on the provider's hands. hands. They are on the chart, on the door handles, on the stethoscope, the chairs in the waiting rooms, the table with the flimsy paper, the instruments, all over the patient sign-out area, etc. They are, in effect, unavoidable. And your child is not the only person who suffers. Every time a doctor who sees children moves to a new area of the country, they are repeatedly sick for a couple of years. It's called "pedi-crud", and they get it from your kids.
That's my take as a health care provider. What can you do?
In every patient room with a sink, there are handwashing instructions posted on the wall. Before you get started into any conversation, notice the instructions and ask the doctor if he/she believes handwashing is critical to fighting the spread of disease. They will probably answer, "Yes, of course."
Well done, you've painted him/her into a corner.
Then answer his/her questions. If before the exam the doctor doesn't move to wash his/her hands, just say, "Um, would you be so kind as to wash your hands? You've convinced me of it's importance." He/she should be quite happy to chuckle and do so. (Most doctors are happy to make their patients happy. It really is that simple.)
If they do not, you may have either a very pragmatic doctor or one without empathy, but it's hard to tell. I would bring the lack of respect for the handwashing request to the attention of the office manager in this case.
I have been brought up short by a patient or two (though not for hygiene); it's a humbling situation but not irreparable. Patients are often very good teachers as well. A wise doctor will listen to what they have to say. Both parties win.