The protection extends until October 1, 2024 and does not depend on when the vaccine becomes fully approved by the FDA.
The official notice of Declaration Under the Public Readiness and Emergency Preparedness Act for Medical Countermeasures Against COVID–19 (PDF) was published in the Federal Register Vol. 85, No. 52 for Tuesday, March 17, 2020, on pages 15198 through 15203. The Declaration was effective as of February 4, 2020.
Section XII of the notice (on p. 15201) reads:
The Secretary must identify, for each Covered Countermeasure, the period or periods during which liability immunity is in effect, designated by dates, milestones, or other description of events, including factors specified in the PREP Act. Section XII of the Declaration extends the effective period for different means of distribution of Covered Countermeasures through October 1, 2024.
Section VI (on p. 15199) reads, in relevant part:
To be a Covered Countermeasure, qualified pandemic or epidemic products or security countermeasures also must be approved or cleared under the FD&C Act; licensed under the PHS Act; or authorized for emergency use under Sections 564, 564A, or 564B of the FD&C Act.
A qualified pandemic or epidemic product also may be a Covered Countermeasure when it is subject to an exemption (that is, it is permitted to be used under an Investigational Drug Application or an Investigational Device Exemption) under the FD&C Act and is the object of research for possible use for diagnosis, mitigation, prevention, treatment, or cure, or to limit harm of a pandemic or epidemic or serious or life-threatening condition caused by such a drug or device.
The news story linked in the question includes the statement:
The protection lasts until 2024.
The pulmonologist is incorrect.