Taking these in order, and starting with the pandit because it's colouring the rest.
I can see what he was looking at. Your 12th house is the foreign residence house and it has 20 bindus, the lowest of any house in your chart, with your Saturn debilitated sitting in it. Looking at that alone, anyone would say foreign settlement is weak.
What it misses is the 12th lord. Mars rules your 12th and it sits in your 10th, which is a kendra and has 35 bindus, second highest in your chart. Mars is also the exact planet that cancels that debilitated Saturn, since the dispositor of a debilitated planet placed in a kendra gives neecha bhanga.
So your 12th house is weak but your 12th lord is strong and it's sitting in your career house. That reads as: you don't get to stay abroad passively, you get there through work, and a first attempt that doesn't hold is precisely what this configuration looks like. Which is what happened to you. Your permit expiring wasn't a sign that India is your destiny. It was a weak 12th house with a strong 12th lord doing exactly what it does.
On timing, you're in Rahu mahadasha till Aug 2028, currently Rahu-Moon till 20 July 2027. Rahu-Moon is a paperwork and preparation period, not an arrival one, and it's also why you feel confused right now. Rahu with the Moon manufactures worst cases in your head.
Your move window is Rahu-Mars, 20 July 2027 to 8 Aug 2028. Mars is your 12th lord sitting in your 10th. That's the relocation sub-period. Then Jupiter mahadasha starts 8 Aug 2028 and runs 16 years, and your Jupiter sits in your 12th house. A sixteen year period of a planet placed in your foreign residence house answers the settling question.
On pathway, your 12th lord is in your 10th, not your 4th or 9th. So it runs through your occupation. Employment based, not study, not family. Your 10th at 35 against your 12th at 20 says the route anchored to your job carries you further than a residence application standing on its own.
Now the IELTS. January 2027 is a good pick. You'll be in Rahu-Moon-Mercury, 5 Dec 2026 to 21 Feb 2027. Mercury is your 2nd lord, the house of speech, and your 5th lord, the house of exams, and it sits in your 9th, foreign travel. It's also in Sravana, the nakshatra of listening. For a language test with a listening module that's about as apt as it gets.
Sit it before 21 Feb 2027. After that the Ketu sub-sub-period starts and Ketu is the worst of them for exams, it scatters focus.
For the day itself I'm not going to invent one without casting it. Pick by this against a panchang. Prefer a Wednesday, Mercury's day and Mercury is your exam and speech lord. Prefer a day the transit Moon is in Taurus, Cancer, Virgo, Aquarius or Pisces, your 1st, 3rd, 5th, 10th and 11th. Avoid Libra, Sagittarius and Aries, your 6th, 8th and 12th. If you can land on Hasta, that's your own birth star sitting in your 5th, take that one.
15 January is fine if it clears those. If not, anywhere up to 21 Feb.
Short answer: Canada, through work, landing late 2027 into 2028.