Mary Stuart had about the same amount of claim against Elizabeth Tudor, so it isn't like we never saw a version of this happen OTL.
Mary and the Lennox Stewarts' claim against Catherine Grey would be more solid than her claim against Elizabeth Tudor, as there are two Margaret Tudor descendants on the line between Mary and Catherine in this TL(and Catherine was 7th in line in 1558 in OTL).
Since a majority of English people in 1558 were either Catholic or nominally Catholic, Papal endorsement for either of the two major descendants of Margaret Tudor would boost support for either Mary or the Lennox Stewarts. No good Catholic would like to find himself/herself supporting a candidate denounced by the Pope as an usurper. Catherine's main support base would be Protestants(especially Southeastern English ones).
Talking about Lennox Stewarts' ability to enforce their claim, while Mary could co-opt the Lennox Stewarts by restoring the title of Earl of Lennox and all of the Scottish holdings of Matthew Stewart, Earl of Lennox, to Lord Darnley, as well as removing the Earl of Arran and his extended family from the line of Scottish succession(a major goal of Lennox Stuarts over years), Habsburgs would still use the Lennox Stewarts as figureheads. This is the exact meaning of being a claimant, which doesn't mean that one actually
claims a throne, but means that someone can
use one to claim a throne. Or, in worst-case scenario, Philip II could claim the throne outright.
The overall foreign and domestic endorsement of English claimants would be like
France would endorse Mary Queen of Scots, of course.
The Habsburg Monarchy would endorse Lord Darnley. If the Lennox Stewarts capitulated to Mary, then Philip II might claim the throne by himself as the heir of John of Gaunt through both Blanche of Lancaster and Constanza of Castile. (Lady Catherine Grey was descended from Blanche's second daughter, while Philip II was descended from Blanche's first daughter).
Domestic Protestants would endorse Catherine Grey.
Domestic Catholics would be in disarray, although they are a majority. Some would back the candidate backed by the Pope. Others would stick to one of the three Catholic claimants. Margaret Clifford would jump out and get some support from surrounding Catholics, but she would be the weakest of the three claimants, as she was the most junior one among the three major Catholic candidates. Lord Darnley would theoretically be the main candidate, but in case of Lennox capitulation, the subsequent actions of English Catholics would be unclear. Either endorsing Mary in exchange for separating the French crown from the British crowns after Mary died or turning toward Margaret Clifford would be possible.
Foreign Protestants would also endorse Catherine Grey, as English emigres in Geneva would likely want to avenge for the execution of Lady Jane Grey.
Some additional notes:
Margaret Clifford's father-in-law was the Lord of Mann, which was close to Mary's home territory. A Scottish occupation of the Isle of Mann(or, in the eye of Marie de Guise and Mary Queen of Scots, "confiscation") would likely discourage Margaret Clifford from claiming a throne for which she was only 6th in line per primogeniture even after the elimination of two major candidates, the Countess of Lennox and Duchess of Suffolk, in this TL.
In any scenario of Catherine Grey victory, Catholicism would be equivalent to treason, and there would be a return to 1552.