I was able to resolve this issue using the above solution. To add more details, I was able to obtain the CUDA version number to use by running nvidia-smi in BASH, which was 11.4 in my case. Then, I did
julia> using CUDA
julia> CUDA.set_runtime_version!(v"11.4")
and restarted REPL. Then the subsequent use of CUDA downloaded the CUDA runtime:
julia> using CUDA
[ Info: Precompiling CUDA [052768ef-5323-5732-b1bb-66c8b64840ba]
Downloaded artifact: CUDA_Runtime
Then I was able to use CUDA normally. Hope this helps people experiencing similar issues.
UPDATE. My system admin says the CUDA version shown in nvidia-smi just means that nvidia-smi program is built with that CUDA version. He says all the CUDA versions available on the system can be obtained by module avail cuda. In my case this showed 11.6 as the most recent version instead of 11.4.