I can't comment on the particular policies of the UK (as I'm not fully aware of them), but in my honest opinion... No.
Regardless of what height you are. I don't think any public/governmental entity, which is supported through public taxes, should offer/support cosmetic procedures. I hold this stance for all of them, even for transgender sex change, as harsh as that sounds. Also, when it comes to "height neurosis," less so.
Mainly because so many things can easily go wrong during the journey. Sure, it's not common, but if during the lengthening, or consolidation phase you fall and cause further damage to your body. You'll be looking into further expense to recover from that. And I sure as fk wouldn't want any more of my tax money going to fixing a fk up for a completely unnecessary procedure that wasn't really life or death to begin with...
It's like there a hundreds of things that money could be better used for...
Plus, I hold it as right of way for someone to work hard for the money needed to undergo LL; at least in some sense you're very active and aware as to what you're getting yourself into...