How evolutionarily conserved are UTRs?
This is a question specific to the organism, assembly, and genes of interest to you. You can answer this yourself with a bit of elbow grease.

You could measure mean phyloP or phastCons conservation signal over UTR-annotated intervals, and compare it against the background mean conservation signal over random, mappable regions of the genome of similar length (i.e., calculate the mean and variance in UTR length and sample mappable regions of the genome for intervals with the same distribution).

At a very basic level, you could perhaps apply either the U-test or Wilcoxin rank-sum test, depending on your assumptions of dependence, whether your background does or does not include UTRs.

One source of phyloP and phastCons signals for various genomes and multi-genome alignments is UCSC Goldenpath. For example, here are phyloP 100-way alignments to hg38. You might get UTR annotations from Gencode, assuming you are working with human or mouse. To do interval overlaps and mapping/statistics operations, you might use gff2bed, bedops, and bedmap in the BEDOPS toolkit(*), and then use R to do statistical tests.

Disclaimer: I am one of the authors of the BEDOPS toolkit.