Several other players - including England pair Finn and Owais Shah - also struggled to make it to the ground on time, although it was only Roland-Jones who actually missed the start.
Worcestershire, who won the toss, had began well before Roland-Jones' arrival, and 47 from Daryl Mitchell helped them to 101-1 with relative ease.
But the opener became the late arrival's first wicket of the day and Worcestershire stumbled to 126-4.
Moeen's 66 from 88 balls included ten fours and a six and pushed the visitors to their first batting point, before he was caught by Shah off Dexter.
James Cameron hit the second six of the innings as he swung Dexter's medium pace over mid-wicket, but went to the new ball for 28, lbw to Pedro Collins as he fell across his stumps.
Gareth Andrew helped Worcestershire to a third batting point with 36 off 31 balls before becoming Roland-Jones' third wicket, and the same man wrapped the innings up with the dismissal of Alan Richardson.
BBC Hereford and Worcester's Trevor Owens: "Worcestershire will be reasonably happy with their first day after the last pair of Jack Shantry and Alan Richardson took them past 300 for the extra batting bonus point.
"However, they had six batsmen score 28 or more, and yet Moeen Ali was the only one to go past his half-century and this will be a frustration to director of cricket Steve Rhodes.
"But they will feel if they can take quick wickets in the morning session on day two then they will put themselves in a great position."
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