Well, as I recall, she gave orders for Robert Dudley to be made Lord Protector of the Realm in the event of her death.
At this point, the heir according to the Will of Henry VIII is Catherine Grey, who been recently removed to house arrest in London for the Seymour affair at this point IIRC. If she succeeds, the most likely outcome is for the priest who witnessed her wedding to the Earl of Hereford to be "found" and the annulment of their marriage reversed earlier than in OTL.
However, there is also Lord Darnley to consider, as he is still in England and not yet a candidate for the hand of Mary Stuart. Despite being excluded by Henry VIII's will, he has a lot going for him by virtue of his sex. His mother, Margaret Douglas, was very ambitious and closely linked with the Catholic faction at court. She's not going to hesitate to put her son on the throne if she has the chance, and considering his rivals are Catherine Grey (a woman who has already disgraced herself and carries a lot of baggage with her connexions to the Seymours and Lady Jane) and her sister Mary Grey (who was deformed and never really taken seriously), Margaret Douglas may well succeed.
If Henry Stuart manages to secure the throne, his mother and the Catholic faction will rule through him no doubt, considering his character in OTL. Remember, at this point, England is still nominally in communion with Rome (Elizabeth wasn't excommunicated until 1570), so at the very least, the Church will probably pursue a conservative course similar to the previous Henrician settlement. Without the Elizabethan settlement, the reformist undercurrent that exists (still a minority, but a vocal and well placed one) is going to be alienated and this will create a lot of religious turmoil--though I can't see it going as far as the Marian persecutions, considering how unpopular it was with both Catholics and Protestants.
Also, the old Marian power structure and policy would likely see a resurgence under Darnley. At the very least, England will be allied closely with Spain, since Mary of Scots is probably still going to claim the Crown (though she isn't really a threat at this point).
If Catherine Grey manages to secure power, she'll likely be dominated by her husband, the Earl of Hereford, considering the consensus seems to be that she wasn't the strongest of characters. This means a possible return to the Edwardian policies and religious settlement (the Seymours were staunch Protestants), which will please the reformists but alienate the Catholics and likely cause trouble down the road as well.
In either case, things will be very different in the coming decades.