There is a strengthening of the Boone-Higman result, due to Thompson. He showed that we can take the simple group to be finitely generated. In full, this reads:
"A finitely presented group has solvable word problem if and only if it can be embedded in a finitely generated simple group that can be embedded in a finitely presented group."
You can find the full details in:
R. J. Thompson, "Embeddings into finitely generated simple groups which preserve the word problem", Word Problems II: The Oxford Book, Studies in Logic and the Foundations of Mathematics, Volume 95, (1980).
As far as I am aware, your original question "Does every finitely presented group with soluble word problem embed in a finitely presented simple group?" is still an open problem.