"Okay, we definitely made improvements to the home say $25k. How does this impact the basis? "
Add to each person's original ($150K) basis the amount that that person paid for improvements. So Mary's basis for her original inherited 1/3 share will be $150K plus the amount that Mary paid for improvements. Then recalculate Joe's and Peter's basis for the gifted portions at the time of the gifts.
Joe's basis at the time of the gift will be $150K plus the amount that Joe paid for improvements. That becomes Mary's basis for the 1/3 share she gets as a gift from Joe.
Peter's basis at the time of the gift will be $150K plus the amount that Peter paid for improvements. Peter's basis for the share that he gifts to Mary will be (50/175) x ($150K + the amount that Peter paid for improvements). That becomes Mary's basis for the share she receives as a gift from Peter.
However, if adding the cost of improvements makes anyone's basis more than $175K, the FMV at the time of the gift, then it gets more complicated, and the basis might depend on the eventual selling price. See "FMV Less Than Donor's Adjusted Basis" in IRS Publication 551, Basis of Assets.
"what will Mary be obligated to pay for capital gains?"
You can calculate your capital gain based on an assumption about the selling price, but in order to calculate how much tax you will have to pay on the gain you would have to know the amount of your other income for the year of the sale and your filing status for that year. The amount of tax would also depend on things that you have no way of knowing now for a future year: what the inflation adjustments of the tax brackets will be, whether there will be any other changes in the tax rates, and whether there will be any changes in the applicable tax laws.
All of the gain will be long-term, no matter when you sell the house, because it was all acquired by inheritance.
