The finiteness of the class group, in schematic terms, means that if $K$ is a number field, then the Picard group of $\operatorname{Spec}\mathscr{O}_K$ is finite. I heard that it is true in general that the Picard group of a projective scheme regular over $\mathbb{Z}$ is finite and that it follows from Mordell-Weil.
I looked for this result in a number of books but found nothing. How could we prove this and what would be a reference for reading about it?
Question: "I looked for this result in a number of books but found nothing. How could we prove this and what would be a reference for reading about it?"
Answer: If $C$ is a non-singular curve over an algebraically closed field $k$ it follows the grothendieck group $K_0(C)\cong \mathbb{Z}\oplus Pic(C)$ is determined by its Picard group $Pic(C)$. If $\pi:X \rightarrow Spec(\mathbb{Z})$ is a regular scheme of finite type over $\mathbb{Z}$, the K-theory $K_m(X)$ is conjectured to be finitely generated. Hence $Pic(X)$ is conjectured to be finitely generated. The Bass conjecture is known for $\mathcal{O}_K$ with $K$ a number field an any curve $C$ over a finite field by the work of Quillen.
There is the following theorem:
If $S:=Spec(R)$ with $R$ a discrete valuation ring and $X$ a regular flat projective curve over $S$ with geometrically reduced and irreducible fibers, it follows $Pic^0_{X/S}$ is a Neron model of the generic fiber $X_K$. Ie if $J_K$ is the Jacobian of $X_K$ its Neron model is $Pic^0_{X/S}$.
Im unsure if a similar result holds for any projective regular scheme over the integers. This site is propably not the right place for such questions - maybe you should ask your advisor or write an email to an expert. The "mathoverflow" site is a site for research questions.
To my knowledge the general conjecture (even for $Pic(X)$) is open. I believe you find some updated references on the wikipedia page:
https://en.wikipedia.org/wiki/Bass_conjecture
https://mathoverflow.net/search?q=bass+conjecture
https://mathoverflow.net/questions/170347/a-weak-version-of-bass-conjecture
It appears from the above posts that the question if $K_0(A)$ is finitely generated for a finitely generated regular $\mathbb{Z}$-algebra $A$ which is a UFD is open.
If you read one of the references you will find that the Bass conjecture is one of the main open problems in algebraic K-theory - the natural thing to do is to ask someone that has published in this area - why not write to one of the authors of the book(s) and ask for a reference? You will find their emails online (some of the papers are available online).