The 19-year-old also claimed a catch as Gloucestershire mustered a disappointing 105 for six in their 20 overs before the weather scuppered Middlesex's hopes of swapping places with their opponents at the foot of the South Group.
Gloucestershire slumped to 37 for four before Benny Howell and Will Gidman restored some pride by combining to add 51 off 46 balls.
Having elected to bat, the visitors started disastrously, failing to find the boundary until the seventh over, by which time they were already three down.
Alex Gidman was caught at deep midwicket without scoring off James Harris, who struck again when Chris Dent found Podmore just behind square.
With Hamish Marshall dragging Podmore on to his stumps, Gloucestershire were 16 for three in the fifth over.
Ian Cockbain pulled Ravi Patel's first delivery for four, only to pick out Joe Denly on the deep midwicket rope having made 20 three overs later. At the halfway stage, the visitors were a mere 39 for four.
Howell swept Patel from outside off stump to the long-on boundary, however, and 14 runs came in an over from Neil Dexter, the 15th, which saw Howell clear the fence at midwicket for six and Will Gidman guide one down to deep fine-leg.
Howell, Gloucestershire's top-scorer with 39, was handed a life on 33 when he was caught in the deep off a no-ball from Harris in the 16th over.
Having just had Howell caught on the straight boundary by Dan Christian, Podmore was handed a third wicket when debutant Adam Rouse was bowled around his legs, attempting an optimistic scoop to fine-leg.
— Middlesex Cricket (@Middlesex_CCC) June 26, 2014
Will Gidman finished with an unbeaten 24 from 23 deliveries, including two boundaries, the best of which was a reverse-sweep through square-leg off Ollie Rayner in the 12th over.
"We bowled tremendously well up front and we would have definitely backed ourselves to win tonight, but you can't control the weather," Podmore said.
"It's a shame we did not get to bat at all. My first game at Lord's - you can't ask for much more from a personal angle, and it's a dream come true really. It's what you work hard for."
Gloucestershire's captain Alex Gidman said: "We were pretty confident that the pitch was only going to get harder to bat on and so, with that in mind, I was clear I wanted to bat first.
"It's very hard to comment because it's only halfway through the game. 105 really doesn't look much at all, but I think we would have had a really good chance still in the conditions."