Some leaf yellowing and shed is normal for broadleaf evergreens like Cherrylaurel in spring, in addition to stress caused by transplanting. The oldest leaves are those first jettisoned - they will be among the lowest and/or innermost on the shrub. Shedding should slow and effectively cease as we move into summer, so long as the plants receive adequate watering when needed.

We do not see signs of pests, and the only disease whose symptoms we see are (old) "shot hole" holes, a very common ailment of both cherrylaurel and their kin flowering cherry trees. Preventative treatment is challenging and not often completely effective; curative treatments do not exist. Like most diseases, severity can wax and wane with the weather, with wetter springs promoting infection and drier weather suppressing it. Avoid having the foliage wet with sprinklers (if they're needed, use them early in the day) so the foliage stays dry, excluding rainfall of course.

Shot hole a very, very common disease that even nurseries with good sanitation and preventative-spray programs still contract. Fortunately, plants usually live fairly normal and functional lives with low levels of infection, since high levels are less common in home landscapes. Whatever infection levels they begin with upon planting usually peter-out to minimal levels over time.

The browning here looks like signs of mild winterburn, a phenomenon that causes tissue damage in evergreens when soil moisture in winter is either lacking or frozen and inaccessible to the plant. Damaged leaves - both from this and former shot hole infection - will be shed by the plant as it produces new growth, either this spring or next spring.

Here is some more troubleshooting and care information that may be useful. For now, no actions outside of routine watering checks (as for any new planting) are needed.

https://marylandgrows.umd.edu/2019/01/07/qa-whats-wrong-with-my-cherry-laurel-shrubs/

https://extension.umd.edu/resource/watering-trees-and-shrubs

Miri

The deer did their own version of pruning when eating the laurels, so they may just need a bit more time to fill back in depending on how much damage was done. Otherwise, yes, trimming does stimulate growth, though make sure the bottom of the plant is wider than the top so it gets enough light (if the top is wider, it can shade the lower stems too much to discourage regrowth). See the "Individual Shrubs" entry on this page and its associated graphic for more guidance: https://extension.umd.edu/resource/pruning-shrubs-and-hedges

Miri