Here's a pipeline you can use:

1. Convert the conservation bigWig (.bw) to Wiggle (.wig) with bigWigToWig, using /dev/fd/1 as a filename for standard output. This should allow you to pipe the result to BEDOPS wig2bed and get a sorted five-column BED file:

   $ bigWigToWig conservation.bw /dev/fd/1 | wig2bed - > conservation.bed5
   

2. Split your intervals up into per-base elements with bedops --chop 1.

   $ bedops --chop 1 intervals.bed > perbase.bed3
   

3. Run bedmap to map your per-base elements to the BED-formatted conservation signal. Normally, unmapped elements report no value. You can use the following command to replace unmapped bases with a score of 0:

   $ bedmap --count --echo --echo-map-score --delim '\t' perbase.bed3 conservation.bed5 | awk '{ if ($1=="0") { print $0"0.0"; } else { print $0; } }' | cut -f2- > answer.bed
   

   This will only make sense if a missing conservation score should be zero. You could alternatively replace 0 with NaN or a similar IEEE-ready placeholder for a non-number.

Finally, if you want to put everything into a one-liner, use file substitutions:

$ bedmap --count --echo --echo-map-score --delim '\t' <(bedops --chop 1 intervals.bed) <(bigWigToWig conservation.bw /dev/fd/1 | wig2bed - ) | awk '{ if ($1=="0") { print $0"0.0"; } else { print $0; } }' | cut -f2- > answer.bed

If you're going to do repeated queries with different sets of intervals, it probably makes sense to do the bigWig-to-BED5 conversion step only once, and drop that in as conservation.bed5 where you need it in mapping.