First link is: https://commons.wikimedia.org/wiki/File:Rembrandt,Christ_before_Pilate(Ecce_Homo),_1634,_National_Gallery,_London.jpg

This question highlights my favorite part of art history: there is no solid answer.

To begin to answer this question, it is important to note that Rembrandt's painting is not a painting in the traditional sense, as in it was not made for the market or for a specific patron. It is a grisaille, a monochrome painting, most often prepared as a study or sketch. This was a method with which Rembrandt (and many, many, many other artists who did grisailles) worked out a composition before finalizing the image in a more complete form. In this grisaille, Rembrandt did indeed include a clock. If you look more closely, you'll notice that the clock itself is unusual. The 6 is positioned at the top, where the 12 should be. This is likely in reference to John 19, which describes this scene and indicates that it took place at the "sixth hour", i.e. noon. By representing the clock in this manner, Rembrandt both indicates that it was the sixth hour of the day and technically 12 pm.

The print that resulted from the grisaille emerged towards the beginning of Rembrandt's career. In the realm of printmaking, Rembrandt was generally a "peintre-graveur", meaning that he both designed and created his prints. This was not the case for much of the prints prior to the seventeenth century (and throughout the seventeenth century - Rubens didn't make his own prints, for example). However, there was a very brief period of time in the early 1630s when Rembrandt collaborated with a printmaker from his hometown of Leiden, Jan van Vliet. Of the reproductive works Van Vliet etched, a few of them are religious in subject; mostly, they are tronies (character studies). If you take some time to look through Rembrandt's own production as an etcher, you'll see very quickly that the style of Christ Before Pilate doesn't quite fit with his hand. For the print in question, Rembrandt passed along the grisaille to Van Vliet, who produced the first state. There are indents on the grisaille that indicate Van Vliet worked directly from the sketch to transfer the composition onto the copper plate. After this first state (version) was produced, Van Vliet sent it to Rembrandt, who refined and changed the details, added shading, and clarified aspects of the composition. Van Vliet finished the figural group in the center and thus was born the second state. The state you have linked to is the fourth of five total for this print. It appears at my brief glance that the changes in states from 2-5 are relatively minor.

To return to your original question regarding the removal of the clock, let's look more closely at the first unfinished state by Van Vliet. There are indications of the clock's structure, done very lightly, which then becomes shaded out in subsequent states of the print. This is where we enter into the realm of interpretation/conjecture with no real answer. Anachronism is not necessarily an issue here. It was extremely common in the early modern period for artists to portray Biblical, mythological, and historical events as if they were taking place in their own day with contemporaneous clothing and architecture. I doubt that is the motivation for the removal of the clock and would rather contend that this is an artistic choice. The composition of the clock with the 6/12 is awkward and confusing. It's an unnecessary distraction in a composition where the focus is meant to remain on Christ and to produce the clock clearly would require more detail than Rembrandt or Van Vliet likely wanted to place on this portion of the background. If you look closely at the shading of the print, you'll notice that even the figures clambering at the base of the column bearing a bust of Caesar are shaded in thoroughly so that they do not cause the eye to stray from the focus of the print, Christ. The inclusion of the clock didn't make the best composition, which was ultimately the goal. Rembrandt chose to engage with a reproductive printmaker such as Van Vliet to highlight his talents and spread his fame beyond his immediate contexts of Leiden and Amsterdam. This took place at the beginning of his career when he hoped to establish himself as an internationally successful artist, in large part through the print market. This print (really, art in general) was extremely carefully composed so as to achieve the high status he sought and, eventually, achieved.

Sources:

Schmuckman, Christiaan, Martin Royalton-Kisch, and Erik Hinterding. Rembrandt and Van Vliet: A Collaboration on Copper. Amsterdam: Museum het Rembrandthuis, 1996.

Zell, Michael. Reframing Rembrandt: Jews and the Christian Image in Seventeenth-Century Amsterdam. Berkeley: University of California Press, 2002.

White, Christopher. Rembrandt as an Etcher: A Study of the Artist at Work. New Haven: Yale University Press, 1999.

Clifton, James D. "Rembrandt's Clock." Notes in the History of Art 5, no. 2 (1986): 20-21.