Nobody disputed Jane's right to the throne, after all, Henry had had his marriage to Anne Boleyn annulled, but there were mutterings about whether he had done away with Anne simply to marry Jane. The Imperial Ambassador, Eustace Chapuys, in reporting Henry VIII's betrothal to Jane Seymour, wrote:-
"Has just been informed, the bearer of this having already mounted, that Mrs. Semel [Seymour] came secretly by river this morning to the King’s lodging, and that the promise and betrothal (desponsacion) was made at 9 o’clock. The King means it to be kept secret till Whitsuntide; but everybody begins already to murmur by suspicion, and several affirm that long before the death of the other there was some arrangement which sounds ill in the ears of the people; who will certainly be displeased at what has been told me, if it be true, viz., that yesterday the King, immediately on receiving news of the decapitation of the putain entered his barge and went to the said Semel, whom he has lodged a mile from him, in a house by the river." L&P x.926
So there obviously was some gossip about the matter but there is no record of any open opposition to Jane and I think she won the people over.