Chelsea Sq. was right down the road from where I lived during my internship! Ah, the memories!

Anyway, to answer your question, I did some digging to see if there were any London street sign history websites that could help in dating the sign. I was a little surprised that nothing popped up specific to street signs, considering all the weird London-related artifact sites that are out there. I had hoped that it would be as straightforward as "X typeface was in use prior to Y date" which would make narrowing down the date of your sign a little easier. Especially since it is a serif typeface, and most modern street signs are sans serif--but that quickly proved a no-go because certain burroughs use serif type in some areas. Also, it appears that sans serif typeface was in use earlier than I had assumed (possibly even as early as the 19th century, though I couldn't turn up any exact dates on the photo in question).

That said, this area did sustain heavy damage during WW2, most notably the Exhibition Road exterior of the Victoria & Albert Museum that still bears shrapnel scars to this day (for reference, the V&A is a little over half a mile from Chelsea Square). Here's a website with an incredible map of bomb sites localized on Chelsea and Kensington, for context.

So, I'd say it's plausible that this is a sign that was damaged in WW2, however, I can't rule out that it was being used as target practice by vandals, either. To get a more definitive answer would require finding photos of street signs from this area between 1930-1986 to compare yours against. It's not impossible, but it would require a bit of patience and a lot of free time.

How did your dad come across it?

Edit: Found a more specific pinpointing of bombings in Chelsea Square. Looks like there were at least 2 that were centered near Chelsea Sq. and several more that were probably within range to do some damage.