It took them several weeks.
Exodus 16:1 They set out from Elim, and all the congregation of the people of Israel came to the wilderness of Sin, which is between Elim and Sinai, on the fifteenth day of the second month after they had departed from the land of Egypt.
Exodus 17:6 Behold, I will stand before you there on the rock at Horeb, and you shall strike the rock, and water shall come out of it, and the people will drink.” And Moses did so, in the sight of the elders of Israel.
Exodus 19:1 On the third new moon after the people of Israel had gone out of the land of Egypt, on that day they came into the wilderness of Sinai.
I cannot think why it should not take this much time. There was no need to get there in a hurry, and we don’t know how quickly they moved, nor how long they camped at each stopping-point.