Lee Porritt in his article The Lyndhurst Portrait was the one who suggested that these two portraits may once have belonged to the same set.
They very much look as if they are the work of the same painter. The composition pattern is the same: «The sitter is placed behind a red cushion or cloth that has been embroidered with the use of gold thread.»
Though not the inscription pattern.
Katherine of Aragon – KATHERINA · VXOR · HENRiCi · Viii ·
Jane Seymour – IANA VXOR̃ HENRiCi OCTAVi ·
The little drawing behind the inscription of the Jane Seymour painting is very similar to the one behind the inscription on the portrait of Mary I Tudor in the Royal Collection. That set is missing its Jane Seymour (presuming it ever had one), but the painting style of that set is very different to this. I therefore wonder if the painting of Mary I Tudor belonging to the set in the Royal Collection could instead be based on the portrait of her that once belonged to this set. RCIN 404739 Mary Tudor has the same composition pattern as these two.
Very similar inscription pattern to this Jane Seymour as well:
Mary I Tudor – MARIA · REGINA · FILLIÆ · HENRiCi · OCTAVi