The (journal) publisher's interest is presumably to comply with conditions imposed by the copyright owner: CUP might demand of Elsevier that their reviewers delete electronic copies of books after the review is finished. In keeping and using the work (and perhaps freely distributing that copy), you have violated CUP's copyright (assume that CUP holds the copyright). Therefore you can be sued. Elsevier can also be sued. What is not clear is how many people can sue you (and whether you can sue anyone).
It would depend on whether you were informed of the deletion requirement before you agreed to review the work. If you were led to believe either that you would get a paper copy of the work, or at least would be able to retain and use the electronic copy for your own use, the journal publisher does not have the option of rewriting the terms of that agreement after the fact. If you reasonably relied on them granting you permission to keep the work, they can't sue you. You might even be able to sue them, insofar as getting a free copy of an expensive book in exchange of writing a review is a standard academic business deal. On the other hand, if they told you in advance that you'd have to delete the review copy, then that is the end of the discussion. If you didn't read the agreement, that too is the end of the discussion.
It is not clear what recourse the rights holder might have against you. You have no contractual relationship with the book publisher, and the journal publisher is not the agent of the book publisher so the book publisher is not bound by the errors of the journal publisher. Because you are expected to know that all IP requires permission of the rights holder to copy and since you know that the journal publisher is not the copyright holder, you are on thin ice in assuming that the copyright holder has granted you permission to copy the work. Indeed, even agreeing to review a work in electronic form is a dubious proposition, without suitable legal assurances that the rights holder has granted permission to make the required copies. If a journal buys a physical copy of a book, it can lend or give it to you to write a review, and no permission is required wrong the book publisher. To review an electronic copy, permission from the book publisher is required. You might try defending yourself against an infringement suit under the fair use doctrine, since the underlying purpose of "fair use" is precisely to allow book reviews.
If the journal publisher was negligent in not informing you of the copyright conditions imposed on them (which they are supposed to impose on you), your infringement may be innocent, and you might only have a small liability. You could sue the journal in case the book publisher sues you – the journal publisher has a duty of care to you.
Technically speaking, you're in trouble once you download the illegal copy, and technically speaking, you infringe copyright every time you read the work. Digital content is, or should be, distributed under some license (a contract between you and the rights holder), otherwise it is illegal to receive or use the work. Hence various public licenses grant permission to copy, subject to various limitations: the use may not be commercial; the work may not be redistributed; the work may not be altered; the work may not be redistributed in exchange for money, or some something of value... The teeth that digital licensing has is that if you copy and use a work contrary to the terms of the license, you do not have permission (the rights holder has granted conditional permission).
Copyright law gives the rights holder the exclusive right to authorize making any copies, and in order to use a computer file, a number of copies are made (by various programs). The basic protection is that "the owner of copyright under this title has the exclusive rights to do and to authorize any of the following: (1) to reproduce the copyrighted work in copies or phonorecords; (2) to prepare derivative works based upon the copyrighted work". Copies are statutorily defined as "material objects, other than phonorecords, in which a work is fixed by any method now known or later developed, and from which the work can be perceived, reproduced, or otherwise communicated, either directly or with the aid of a machine or device. The term “copies” includes the material object, other than a phonorecord, in which the work is first fixed." One or more "derivative works" (with no added creative content) are created in getting from a pdf file to a computer screen.
In §117, Congress created an exception to general copyright protection, whereby one can make "another copy or adaptation" of a computer program provided
Notwithstanding the provisions of section 106, it is not an
infringement for the owner of a copy of a computer program to make or
authorize the making of another copy or adaptation of that computer
(1) that such a new copy or adaptation is created as an essential step
in the utilization of the computer program in conjunction with a
machine and that it is used in no other manner, or
(2) that such new copy or adaptation is for archival purposes only and
that all archival copies are destroyed in the event that continued
possession of the computer program should cease to be rightful.
This exception only covers computer programs, and not data files, and as the Copyright Office says
You are not permitted under section 117 to make a backup copy of other
material on a computer's hard drive, such as other copyrighted works
that have been downloaded (e.g., music, films). It is also important
to check the terms of sale or license agreement of the original copy
of software in case any special conditions have been put in place by
the copyright owner that might affect your ability or right under
section 117 to make a backup copy. There is no other provision in the
Copyright Act that specifically authorizes the making of backup copies
of works other than computer programs even if those works are
distributed as digital copies.
I say that "technically", it is infringement if you open and process a digital file that you do not have permission to copy (via a license), because that is what the law says, and no court case has deemed that Congress intended something entirely different. It is notoriously difficult to establish legislative intent. Since Congress did articulate narrow exceptions in similar cases (computer programs, also secondary transmissions of performances and display of works, and did not include any exception for copies made by computer programs in the course of "using" a protected work, but could have, we can conclude that Congress did not intend to create such an exception. One might point to other facts to argue that Congress did intend that. At present, I believe that the Supreme Court would uphold the letter of the law, but there is really only one way to find out. That is, at any rate, the basis for the copyright holder pursuing an action against you (or, you and the journal).
Even without any prior notification of a requirement to delete, they can still tell you that you must delete the work. They probably could not successfully force you to delete the work in a court of law, but there are many other things that they can do that boil down to the fact that you must delete that copy of the work. Blackballing, for example.