Hi Nanako,
We use "fall through" to describe a plan or endeavour that has not come to fruition and "fall apart" for several other meanings: (i) disintegration of something that was originally intact or whole. (ii) something that is wearing out due to prolonged use (iii) something that was working but is no longer so (iv) being emotionally overwhelmed or incapacitated.
Since your plan for an overseas homestay failed to materialise, I would go with "fall through". However, in my opinion, if you had wanted to dramatise how your travel plans were foiled due to a string of bad luck or unexpected mishaps of epic proportions, you could have used "fall apart" instead. Let's exaggerate: You have planned the perfect homestay, but you missed your flight because you mistook the "12 a.m." departure time for "12 p.m." You rushed to get another ticket over the airline counter at an exorbitant price and the next plane took off, only to make an emergency landing an hour later due to an engine failure. When you finally reached your destination, you were denied entry and turned back as your passport did not meet the minimum expiry requirement. Your plans have really "fallen apart".
Your question reminded me of a literature book that my school prescribed when I was a secondary school student. The title of the book is "Things Fall Apart", written by a Nigerian author, Chinua Achebe. In the story, a bright student from Africa had studied overseas and returned home with many ideals; however, his future "fell apart" when he succumbed to pressure from his countrymen and world pleasures and was eventually arrested and brought to court.
I hope this helps.