How can one make up for a year's worth of missed prayers due to the unavailability of a place for ritual bathing (ghusl) and embarrassment from performing ghusl in front of her mother-in-law?

What the questioner mentioned is not an excuse that permits abandoning prayer, and she must repent for her negligence.

According to the majority of scholars, missed prayers must be made up. This is the more cautious approach to absolve oneself.

The make-up is done according to one's ability, in a way that does not harm the body or livelihood.

Order should be observed when making up prayers, to avoid disagreement, even if it is not obligatory according to us.