It is unclear whether or not the king was at Jane's side when she passed away - as The Tudors shows in a deeply moving and heart-wrenching scene during the third season. However, we do know that Henry was extremely upset and grief-stricken after his beloved wife's death. In his mind, Jane was his only true wife. She had done everything a royal wife was supposed to do at the time - She had listened to him, turned a blind eye when he flirted with other women, delivered him the precious son he had waited for, and had never defied or betrayed him in any way. Now, because she was dying after fulfilling Henry's greatest wish, she was in all ways the perfect Queen.
We can even see the grief that Henry felt in a letter he sent to King Francis I of France soon after Jane's death:
"Divine Providence...hath mingled my joy with bitterness of the death of her who brought me this happiness."
Such a sad day in Tudor history. The people loved their Queen, the King loved his Queen, and now the infant Prince was without a mother. However, Jane's legacy and love would live on, and she would be the only one of Henry's wives to receive an elaborate funeral fit for a Queen. Henry would certainly never forget the woman who had brought him everything he'd desired, and whom he considered to be his true wife.
Rest in Peace, Queen Jane Seymour!