This would have to be
one of the most opinion-based questions we could possibly have, and
one of the most important to deal with in a sensible manner.
Firstly, it must be said that what is safe depends on the woman's own behavioral history, and the state of the pregnancy. She certainly should not be taking up new active sports.
Some competitive sports ban obviously pregnant women from participating, not through any risk to the woman or child, but because she has two advantages. She has a greater blood supply, and competitors are likely to "go easy". I recall this was the case when I played competitive squash.
The mother of my children cycled (on a road bike) until month seven in both pregnancies. With her extra blood supply, she would leave me behind on the hills! She only stopped when her belly was hitting the top bar. It should be said that both pregnancies were completely normal. She did discuss cycling with her Ob, who agreed that since she was already a cyclist, she could continue, but that she should be sensible and stop if problems developed. She cycled about 60km (38 mi) in 3 or so rides per week.
She did find her balance changed. By doing the activity continuously, she adapted to that.
My recommendations are
Listen to the obstetrician. You chose this person for their guidance, for a reason.
Only cycle if already a regular, confident cyclist.
Always ride in familiar, safe places, and with a buddy.
Ignore those who use emotional blackmail in their arguments. This is an emotive topic; it needs clear factual arguments to form a balanced decision.
Every activity involves risk, including no activity, and being wrapped in cotton wool.
PS The great Sheldon Brown's alter ego Carapace Completed Umber also wrote about Riding while pregnant.