Ok, let's start from the beginning here. Lots of info with lots of assumptions made all leading to the questions not really being answered.
"Short" answers:
At this point, assuming the drummer doesn't want the song played, can the band play the song? Do royalties go to the drummer?
The original drummer ("OD") cannot stop the band from playing the song. OD cannot stop the band from recording the song (but OD will receive a share of the mechanical royalty). OD cannot stop the band from licensing the song for use in movies & tv, etc (but OD will receive a share of the synchronization royalty).
what about the new one? I assume he's under the legal entity of the band. Does he get royalties?
The new drummer ("ND") will not acquire a share in the copyright to the composition. ND will, however, be entitled to royalties for licensed uses of the recordings ND played on unless the band had a different agreement (such as paying ND for playing on the recording in lieu of a share of the copyright).
Long answer: First, I'm going to assume the question is being asked about the United States. There is a treaty in place (the Berne Convention) that the signatory countries are expected to adhere to and have their own laws meet certain minimum requirements, but not all countries' laws are ultimately the same, even among the signatories.
In the States (and possibly elsewhere, but I've only studied the States), that's colloquially known as a "song" is actually split into two categories: the composition (frequently simply called the "song") and the performance/recording.
Songwriters initially own the copyright to the composition. Performers initially own the copyright to the recorded performance.
Royalties get paid based on the usage of the song and the recording. In most instances, these usages require a contract between the copyright holder and the music user.
What you've seen people refer to as the "mechanical" rights is an exception to the contract requirement as the copyright statutes handle the "mechanical" reproduction with what is known as a "compulsory license." Basically, once a song has been recorded and released to the public, anyone can record that same song and release it to the public provided they follow the requirements of the law (which involves accounting to the songwriters and paying a licensing fee that is set by statute - i.e. the "statutory rate"). There are agencies that have been set up as basically copyright clearance houses that will license the mechanical rights for nearly any song for the statutory rate and deal with the copyright holders itself so the users don't need to.
Anyway, copyright rights can be contracted away, and often are when you start talking about major label recording artists and those who write songs for them. Major labels will insist that bands assign the copyrights in the "masters" (the recordings) to the label. These rights will revert to the performers after a period of time, I believe around 25 years.
Almost all songwriters/bands signed to a label will contract with a Music Publisher to administer their copyright. There are many forms these contracts take depending on the artist's clout and on whether the publishing company is expected to seek out licensing opportunities for the songs, or merely collect and distribute the royalties.
It starts getting complicated here with discussing of the "writer's share" and the "publisher's share," but I think that goes beyond the scope of OP's questions.
Without a contract specifying otherwise (written or oral, but written is better in case of disputes), anyone who contributes to the writing of a song has an equal share of the copyright. This is how you wind up with some songs that have 20+ people listed as songwriters in the credits.
Being part of the band has nothing to do with whether you have a piece of the copyright to the composition. You can help write the song, leave the band before it's recorded, and you still own part of that song. Band members that replace you do not take over your share of the copyright.