I've never seen any SOP where this is a stated procedure nor is it likely to be reliable because of the update rate and the changing variables.
It doesn't have to be an SOP to be useful. Just another piece of information, another clue to what might
really be happening around you.
I've seen wind speeds of 100 kt and varying wildly whilst taxiing and seen the ground speed calculated to be anything between 5 and 20 kt while we were parked at the gate!
Hmmm... What are you flying?
With the conservative values used for performance calculations and safety margins, I think one would notice any relevant change in the wind conditions especially when ATC say "Wind is xxx, you are cleared for take-off."
I take it you've never operated in Africa or the Caribbean then. "Wind calm" can mean up to 25kts from any direction and no, there isn't always a windsock.
On all the modern jets I've been on the wind readout seemed pretty accurate and instantaneous. If I was on the takeoff roll in some dark corner of the world and the displayed wind component was much more adverse than I thought it should be, I would check the IAS against the GS and
consider a reject at low speed or 'firewalling' if at high speed. You're not going to get a windshear warning from a steady (or slowly increasing) tailwind, even though it might put you in the trees at the end of the runway...