it depends on the region and how busy the court is.
Again, let me do the math. Opening a file in court usually costs 120 dollars, with very few chances to win. But waste you a couple more months. Plus the time you will be waiting again on 2nd HC. Without a work permit, how long your finance will last?
Or talking to a new lawyer, it will be another one or two weeks. He or she will give you a better suggestion for stopping deportation. Or, once your 2nd file is received by IRCC, and they give you AOR. You will be safe till the result whether is positive or negative.
Prra is also on the table.
If i were you, i would start to search online for a list of lawyer, and call a lawyer asking for an appointment ASAP. Before doing that, zero thought on judicial review. Because it will consume too much time. Plus you have children to raise.
Make plan A B C D, and prioritize them from the best move to the least best move.