How accurate are doctors in knowing the date of conception once a woman has had an ultrasound?

Depends. +/-3-5 days by ultrasound before 10 weeks. +/-5-7 days by ultrasound between 10-14 weeks. +/-7-10 days by ultrasound between 14-20 weeks. +/-10-14 days by ultrasound between 20-24 weeks. +/-14-21 days after 25 weeks.
Within 3 days . If the ultrasound is done in the 1st trimester it is very accurate in dating the pregnancy usually within 3 days of accuracy. The later in pregnancy the ultrasound is done, the larger the windows of accuracy of the ultrasound.