Luke 1:56 says that she stayed with Elizabeth for 3 months. We know for certain, then, that Mary was at least 3 months pregnant with Jesus. She went to Elizabeth when Elizabeth was in her 6th month of pregnancy (Luke 1:36; 1:39). The NIV says that Mary left immediately and stayed for 3 months. The trip from Nazareth to the town that Elizabeth lived in was a week apart. So we know that, unless she conceived at another time other when the angel talked to her, she was in her 3rd and 1/2 month when she left.
The NIV (New International Version) tells us that at that time Mary got ready and hurried to Elizabeth (Luke 1:39), suggesting she left ASAP.
It is very likely, although irrelevant, that Mary was able to witness the birth of John the Baptist. (She may have been not too excited about giving birth to Jesus) She went to Elizabeth when Elizabeth was at least 6 months along in her pregnancy (Luke 1:36). Mary stayed 3 months (Luke 1:56), so she most likely saw the birth of John the Baptist.
If she left immediately to Elizabeth, when she came back it is unknown to me if it would have been obvious. The Bible gives no indication that she was obviously pregnant.
Mary was at least 3 1/2 months along (Luke 1:56). You said in your question that the trip was about a week away; so she traveled for a week, stayed for 3 months and traveled home for another week. This would actually total to 3 1/2 months. Therefore Mary had to have been at least 3 1/2 months along in her pregnancy by the time she returned to Nazareth.