Firstly the council had pretty much decided that they would proclaim Catherine Grey or failing her they would opt for a domestic nobleman (probably Huntington who had a loose claim through his Plantagenet descent)
Catherine Grey's advantage was that legally she was heiress presumptive under statute. Her disadvantage was that she had married without consent and the Queen had it declared invalid making her son illegitimate - there was a solution a miraculous witness and the Archbishop says all is well.
The claimants:
Under Henry VIII's will
Lady Catherine Grey, Countess of Hertford and her son (she is also probably pregnant with her second at this point)
Lady Mary Grey
Margaret Clifford, Countess of Derby and her sons.
Under Primogeniture
Mary Queen of Scots
Margaret Douglas Countess of Lennox
Henry Lord Darnley
Charles Stuart
Then the Grey's etc
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At this period - the Elizabethan Settlement (in terms of the church) had only narrowly got through and while restoring the royal supremacy it had included many aspects of Catholicism that people would recognize and accept. However, vast swathes of the country were still Catholic, but you also don't have the decades of resentment of an imposed Protestantism that appeared later in Elizabeth's reign. Also at this point she hasn't even been excommunicated by the Pope yet. Also the penalties against Catholics were very light and were not universally or even strictly imposed (as they would in the later reign). The protestant clampdown later was a reaction to the constant and growing Catholic threat to the Queen.
As to Mary Stuart in 1562 she is an unmarried woman, newly returned to Scotland, distrusted by some of her own people - and there is little chance in the short term of outside foreign help for her to take the English throne (and she is unlikely to have the domestic resources to take it by force). English Catholics especially in the north may well support her claim but it's debatable if they have the strength of force to take London and impose her accession.