It appears that the Royal Mint was aware of the constant width problem for £1 coins in slot machines, and unlike the old threepenny bit, has rounded the shape slightly.
“The government acknowledges the concerns over the performance of a coin with a non-constant diameter through the automatic vending process. Testing conducted by The Royal Mint, in partnership with a number of respondents, confirmed that these reservations were well-grounded for a 12-sided coin with flat sides and sharp corners. However, the introduction of rounded edges (‘radial chords’) to the design led to a significant improvement in its rolling behaviour, to a level consistent with that required by existing equipment.”
Consultation Report quoted in http://seanelvidge.com/2016/11/new-pound-coin/