Zacharias, the father of John the Baptist, was of the priestly line of Abijah (Lk 1:5), and while he was serving in the temple, an angel of the Lord told him that his wife, Elisabeth, would conceive. The priesthood was divided into 24 courses, of which the line of Abijah would be the 8th course (1Ch 24). Jewish temple practice was such that each course will serve a week twice a year. Since there are three major feasts at which all Jews must gather in Jerusalem, there are three times in a year when all priests would be needed to serve together.
For a moment, let's just assume this is the first time in the year that Zacharias is serving. The religious new year begins on Nisan 1 (being the "first month" which approximates to March). Zacharias' course would have started on Iyar 27 (in April) - and the following week would be the Festival of Weeks (Shavout). This explains why the "whole multitude of the people were praying" outside -- because the Torah requires all Jews to be in Jerusalem to celebrate Shavout. After the Festival of Weeks, Zacharias returned home and Elisabeth conceived John (Lk 1:23-24). This means John would have been conceived no earlier than the month of Sivan (which is around May).
It is very likely that John was conceived two months later ("after those days") in the middle of Av (about July). Following a 39-40 week pregnancy, John would have been born on Passsover (Nisan 15) -- which is in March. It is important to note that traditionally, Jews expect Elijah to return on Passover -- which is the season of redemption. Christ himself said "Elijah is come already, and they knew him not, but did unto him whatsoever they would" (Matt 17:12, Mk 9:13), thereby confirming John, for he did come in the "spirit and power of Elijah, to turn the hearts of the fathers to the children, and the disobedient to walk in the wisdom of the just; to make ready for the Lord a people prepared for him" (Lk 1:17).
This brings added significance when John proclaims "Behold, the Lamb of God, that takes away the sin of the world". For John was himself born on Passover, and his first introduction of Jesus was as the true sacrifice Lamb of God.
Gabriel appeared to Mary when Elisabeth's pregnancy was into the sixth month (Lk 1:26). This will put Gabriel's visit in the month of Shevat (which would be around January). Mary then hurried to Elisabeth's house, learning that she is pregnant (Lk 1:39). Arriving in January, Elisabeth would have been in her third trimester. Seeing both Zacharias and Elisabeth are old, Mary stayed on to help them. This was especially needful as the baby would be full term around Passover and Zacharias would have to serve in the temple. Mary stayed on with Elisabeth until John was born in March -- thereby staying with Elisabeth for "about three months" (Jan - Mar) before she "returned unto her house" (Lk 1:56).
If Mary conceived when Gabriel visited her (in January), Jesus would have been born in late September / October. More precisely, Jesus would have been full term around the Feast of Tabernacles (Sukkot). All Israel would gather in Jerusalem to celebrate the Feast of Tabernacles to remember how the LORD had brought them out of Egypt and tabernacled among them in their journey to the Promised Land. This explains many things. (1) It is a convenient time to tax people - all Israel will all be gathering in Jerusalem anyway -- no wonder "there was no room for them in the inn" (Lk 2:7) (2) It is not yet too cold - travelling in winter just to get taxed may just trigger another revolt -- the memory of the Hasmoneans was still fresh; (3) the cool (but not freezingly cold) climate also explains why shepherds can still be out tending the sheep.
Immanuel came down on the Feast of Tabernacles to literally tabernacle with Man. God the Father made sure, when Jesus came, the whole of Israel was gathered to honor that day (whether or not they knew the true significance). What more, in Lev 23:36, the LORD declares that during the Feast of Tabernacles, "seven days ye shall offer an offering made by fire unto the LORD: on the eighth day shall be an holy convocation unto you."
All Jewish boys have to be circumcised on the eighth day (Lev 12:3). Therefore, on the day the LORD has declared a "holy convocation", Jesus was circumcised in the Temple.
By the above chronology, everything (logic, common sense, Jewish tradition, and the Law) all begins to converge and make sense, and it is mind-blowing to know that God has ochestrated all these strands together. Indeed His thoughts and His ways are higher.
