The ending of Show Omens is not the ending of Book Omens (or of it's non-existent sequel). There are several reasons that I think make this a fairly solid conclusion:
For one, Book Omens doesn't have a sequel. And it almost certainly was never really going to have one. I mean, it was published in 1990. Terry Pratchett died in 2015. That's 25 years in which they could have written a sequel but chose not to.
In fact, in an interview thing that's conveniently included with my copy of the book, they answer the question "Why is there no sequel?" like this:
Does this sound to you like they have an actual, fleshed out plan for a sequel? It sounds more like they were just throwing ideas around, not even in a drafting phase yet, for a story they didn't really expect to fully write. They also last did this at least 11 years ago, likely much longer.
At best, whatever parts of their ideas made it into the show are the result of one of the two people involved using pieces of something unfinished, likely barely even started, from years ago, without the input of the other author. It certainly can not be a fully accurate representation, simply because one of the people involved is no longer able to add his own insight.
But also, just looking at the book itself, it doesn't really strike me as being written with a sequel in mind. It literally ends with a vision of the future, which is fairly conclusive and also not particularly compatible with the show ending:
This ending does not exactly scream "needs continuation" or "there is further trouble ahead". More the opposite, really.
The scene right before this ending also serves as a very neat bookend and conclusion to the whole story, with Adam stealing an apple and deciding that life would be less fun if people didn't. That eating the apple was worth the trouble you got into for doing so.
The closest we get to foreshadowing a sequel is Crowley mentioning the possibility of the "Big One". But it's not exactly treated as something that is bound to happen. So while I guess it's possible it was actively meant to imply a sequel, the rest of the ending makes it somewhat unlikely imo. More likely it's either incidental or a more vague consideration without an actual, already existing plan attached.
Secondly, it would require a somewhat strange genre and tone shift. Good Omens is a comedy that parodies various things (The Omen, Revelations, The Cold War) to make it's point in a fairly light hearted way. It's universe is somewhat silly but also hopeful.
For it to end the way the show did, it would realistically have to change it's genre. It would have to be a Drama or even Tragedy with a story that seems to have somewhat lost it's parody elements as well as it's hope for the future. Which would be a very strange thing for the sequel to a fairly light hearted and hopeful comedy to do.
It would also be very odd in terms of how Terry Pratchett books usually end. Like, even the more bleak Discworld novels, like Night Watch, do end with the world in tact and at least marginally better than before. There is always hope for the world there. An ending in which the world can not be saved and every single character experiences identity death would be very much out of the norm. (I can't say if it would be for Gaiman, the only other book of his I ever read was Coraline).
Finally, related to this, it is not thematically consistent. Some of the major themes and points of Good Omens are, I would argue:
Both the best and worst of humanity is purely humanity's doing. They have a capacity for great good and evil and either way deserve to exist as they are. Free will and the knowledge of good and evil were worth the cost.
Just because it is written, does not mean it must be done. You can not and should not live your life doing only what someone else has decided you ought to be or do. Your choices, human choices, matter and they make a real difference.
The future may seem bleak, there may be a lot of things wrong, but it is not hopeless and it is worth fighting for. This world and its people are worth fighting for. The only real way to improve it is to get it into people's heads that their actions have consequences and that they should be kinder to each other and the environment.
The show ending is thematically opposed to this. Humanity's will and choices are not the deciding factor, do not make the ultimate difference. The humans of the old universe do not deserve to exist as they are. That imperfect world is not worth saving, can not and should not be saved.
As far as the show seems concerned, Adam, at his worst, was right. The world wasn't salvageable, it's best if we just destroy it and start all over again.
It would be extremely strange to have a sequel contradict the themes of the original like this. Especially since the original themes are ones extremely common in Terry Pratchetts work in general while the shows treatment of humanity, the world and it's future really don't seem to mesh well with them at all.