They were bred from corrupted Elves, which are immortal. But did they retain their immortality after they were corrupted?
So they were not immortal apparently.
I think that it’s important to note that Elves themselves are not “immortal” as the term is used outside Tolkien but “longaeval” – a medieval term not much used in today’s world because there’s little aside from fungal clones which it can be applied to.
An Elf is quite capable of dying – he (or she – I’m going to stick to the common-gender “he” in this post) can be slain, he can die of grief and weariness, presumably he can catch a fatal disease (though Tolkien is very loath to speak of illness as it affects Elves), etc. What he will not do is die of old age. And while the spirit of a deceased Man goes “beyond the circles of the world” to a place only Eru Iluvatar knows, the spirit of an elf goes to the halls of Mandos until he is reincarnated in a new body, ultra-similar to the one he had died in.
This is represented as being a part of the nature of the Elves – and they are jealous of Man’s ability to truly die.
We are, AFAIK, given no clue anywhere in Tolkien’s writings as to what happens to an Orc that is not slain. Nor is there any evidence that an Orc can be redeemed, and join the “good side” in the moral aspects of the combat that runs throughout the Tolkien universe.
Ah, but your quote was from “Morgoth’s Ring” and hence not quite canonical. Call it supracanonical. Part of the canon to be that was not completed.
JRRT wrote himself into a bit of a bind. Initially he wrote that the orcs were bred from elves corrupted by Morgoth’s arts. But later he fervently wished to have them be descended from men. Unfortunately that did NOT fit into the timeline he’d already written, since the orcs appeared first long before Men awoke at Hildorien. Much of his subsequent writing reflected his dilemma. He never was able to reconcile his earlier vs later writings regarding the origin of orcs.
So should we abide by the author’s intent? He clearly wished for orcs to be mortal, and clearly wished that they were descendants of men. Or do we go by the canonical published works, which give an elvish origin, and imply a very long lifetime? (Shagrat chatting with Gorbag about “The Great Siege” which took place 3000 years before).
Personally I think JRRT should have quickly blended his elvish origin orcs with men as soon as they became available, and made them mortal that way. We’d have some orcs of an immortal strain, giving way to far more mortal ones as time went by. But JRRT never asked me. 
Oh, and we can actually determine the minimum age for at least one orc. Bolg. He was the son of Azog, who was slain in 2799 in the Battle of Nanduhurion. So at the latest, Bolg could have been born in 2800. Bolg himself was slain in 2941, at the Battle of the Five Armies. This gives him a minimum age of 141.
Now consider that as the line of Elros waned, it was unusual for even the kings of Numenor to live much past 200 years.
Make of that what you will.
On the other hand, there are indications that it’s very rare for an elf-human pairing to produce young. I find it hard to believe that there have only been three pairings of the Children, and some others are mentioned in the semi-canonical writings (in the Unfinished Tales version of the tale of Turambar, for instance, the strongarm chieftain who enslaves Turin’s mom also has a Noldorin wife). But those three, and possibly another in the ancestry of Dol Amroth, are the only cases we know of half-elf children. Now, admittedly, orcs (whatever they are) breed much more rapidly than do Elves, so it’s conceivable that they’d be more interfertile with humans.
You also have the matter that Saruman’s breeding of Orcs with Humans is regarded as a new innovation, which would obviously not be the case if Orcs had human ancestry. But I suppose that, had Tolkien wanted, he could have re-written that part.
What Tolkien Officially Said About Elf Sex
Here’s a handy-dandy guide to elf on elf relations, along with those rarer elf-human pairings.
Also, orcs make good Cheerleaders!
RAH! RAH! RAH! GOOOO-OO SARUMAN!! YAAAAAYYY!