I've spoke with the Câmara Muncipal in Alvaiázere
Local buses services to the town are operated by Rodoviario do Tejo -
There are only a couple of other services that might help.
The main one is the weekday service from Alvaiázere to Caxarias Estação (Station). the closest station to Fatima.
There are frequent (almost hourly) trains from Caxarias Estação to Lisbon. http://www.cp.pt
It appears that at week-ends there are no regional public transport services serving Alvaiázere.
Alvaiázere to Caxarias Estação (Weekdays only) - Transit Time about 40 minutes.
Departs:
07:05
07:35
11:30
13:40
There is also a weekday service to Tomar.
Alvaiázere to Tomar - Transit Time about 50 minutes.
Departs
07:00
07:20
As you have already appear to have found out Rede Expressos ( http.//rede-expressos.pt ) operate a weekday service from Alvaiázere to Lisbon.
Dept: 06:05
Arr: 08:30
Finally, there is a RodoTejo Rápida service that passes through Valbom, 2,5 km from Alvaiázere, at 17:30 on Saturdays, Sundays and Public Holidays arriving at Lisbon Campo Grande at 19:15.
(Pdf timetable here -
There are also departures on this route on weekdays at 06:30 and at 14:15.
