That query should work just fine in MySQL (>=5.1). MySQL should accept that syntax.
But there could be some differences. By default, Oracle is "case sensitive" and MySQL is not. So one difference you might run into is the number of rows eliminated by the DISTINCT keyword. (You could also encounter this difference between Oracle and Oracle, or MySQL and MySQL.)
Obviously, the tables referenced by the SELECT statement need to exist on the local MySQL database, and the user running the query needs to have SELECT privilege on them. (In Oracle, these references might be references to views or to synonyms, rather than tables. And those views and synonyms might reference objects in other schemas.)
MRCONSO
MRCONSO_UMLS
SnomedCTtoICD10
It's not possible to run a query on MySQL to retrieve data from a remote Oracle database. So, for this query to work, those objects will need exist on the local MySQL database.