Hypothetically the songwriter owns "the song" so will get final say
HOWEVER - in the real world, like every other type of negotiation, the one in the strongest position is the one that has the BATNA - the Best Alternative To a Negotiated Agreement
In this scenario the songwriter has managed to get a publisher to pick up a track, the publisher has managed to secure a cut, the artist and producer are discussing the actual recording
These are 3 MASSIVE steps for any songwriter - even those WITH hits under their belts
Would the songwriter really want to risk going back to square one!?!?!?!?
I think the songwriter knows they wouldn't, the publisher would know this - and most importantly the artist and producer would know this
They would argue that there are 100s of songs they can choose from - and they are probably right
The song would have to be an absolute guaranteed smash hit for the songwriter to have any kind of "batna" and even then a smash hit is no good with nobody to record it
You would also risk alienating yourself and getting known as a "precious" high maintenance song writer - like any industry, the music industry is a small world!
Unfortunately I think you would need to take what you are given at this point - and everyone knows this!
It is nothing new and has been happening since the birth of "popular music"!