Depends on two things - how big the plant is and where you are in the UK. These palms are barely frost hardy if they are more mature, and suffer wind damage quite readily, but temperatures below minus 6degC for a couple of days may kill it, especially if its young.

Actually, that hardiness applies to its cousin W. filifera, the California fan palm, W. robusta is only hardy down to abour -3º C.

https://www.tropicalbritain.co.uk/washingtonia-robusta.html
found this uk link. it suggests winter protection [-5 c seems to be its lower limit]

Also it needs a bright sunny position.