The answer to this requires a bit more information, such as is the house titled in a Trust, and also I would want to read the actual loan papers. But, I can tell you that generally, the Mortgage company did not give you the loan, and has no obligation to you to let you assume the loan. They won't care that you can prove that you have been making the payments.
Having said this, as long as interests rates remain the same or lower than your mother's loan, there may not be an incentive to make you refinance that loan in your name. However, if loan rates are greater, then there is no incentive on their part to allow you to assume a loan that they can get at a higher rate.
I am not sure how the property is titled. If it is in a Trust, and you are the beneficiary and successor trustee, then you just have to get it put in your name. If it is in your Mom's individual name, you will have to open a Probate.
As for putting your wife's name on it, if you own it, you can add anyone you want on it. But, at the moment, it would be considered to be your separate property. I am wishing you a long and happy marriage, but in the case of divorce, you may want to keep it as your separate property until you get a few years into the marriage.