Alex revealed she struggled to stay awake while she was in the car because her son, one-year-old son Kit, was teething.
The Welsh presenter spoke about the situation, which she described as “a new low” in a video on her Instagram story.
Alex, who welcomed her and husband Charlie Thompson’s second son in May last year, was seen shaking her head in dismay as she shared: “Oh my goodness!
“I’ve just been sleeping in the car park in Sainsbury’s, a new low.”
The mother-of-two captioned her post with: “WARNING… Teething babies can lead to…”
The One Show continues today at 7pm on BBC One.