I doubt you will see a remastered "Young Sherlock Holmes" unless Paramount includes it in a future 4k Blu-ray Paramount Presents release. Even then it will be a crap shoot as to whether they DNR'd the hell out of it and/or botched the encode.
I thought the picture looked fine on this disc. The sound on the other hand…
It might just be my TV, but I found the dialogue at a normal level made the music and sound effects deafening. Adjust to the right level for the music and FX and the dialogue was way too quiet.