How does pregnancy affect airline pilots operating under FAA regulations?
As long as the pilot is having a first class medical, her pregnancy should have no effect on the operations, at least legally, though the regulations differ from country to country.

I'm not aware of any FAA regulations that prohibit pilot ing during pregnancy. The closest it comes is in 14 CFR § 61.53 - Prohibition on operations during medical deficiency, which says (noting that pregnancy is not a medical deficiency in any way and the regulation is taken here in the widest sense) that:


(a) Operations that require a medical certificate. Except as provided for in paragraph (b) of this section, no person who holds a medical certificate issued under part 67 of this chapter may act as pilot in command, or in any other capacity as a required pilot flight crewmember, while that person:
(1) Knows or has reason to know of any medical condition that would make the person unable to meet the requirements for the medical certificate necessary for the pilot operation

As long as the pilot feels that she is medically fit and has a valid medical certificate, they can't be legally prevented from flying the aircraft.

In case of other regulattions (like EASA), pregnant women are not allowed to pilot after 26 weeks of pregnance. For example, UK Aviation Authority says:


(c) Pregnancy
(1) In the case of pregnancy, if the AeMC or AME considers that the licence holder is fit to exercise her privileges, he/she shall limit the validity period of the medical certificate to the end of the 26th week of gestation.  After this point, the certificate shall be suspended.  The suspension shall be lifted after full recovery following the end of the pregnancy.
(2) Holders of class 1 medical certificates shall only exercise the privileges of their licences until the 26th week of gestation with an OML.

There are a number of medical issues that the pilot should be aware of while pregnant, so that appropriate measures could be taken, like,

Ability to move the controls to their full extent.

Ability to wear seat belts.

Effect of high altitude operation on the foetus
among others. The airline policy will vary and they have to consider a number of other conditions- for example, the ability of flight crew to egress during a crash etc.