I figure it would work out like this:
Opposing sides square off at say… 600 meters. British in the typical 3-4 line formation of the day (the famous 2-line formation didn’t come about until the Napoleonic wars), and the Romans in their formation.
We’re assuming that the British realize that the Romans are armed with melee weapons and sit back, allowing the Romans to advance.
Romans advance. At roughly 300 meters, the British start firing. Romans are still advancing in step- let’s say a 100 steps per minute pace, for 2.7 mph. That means that they’re covering 79.2 yards per minute, or 1.26 minutes per 100 yards.
So it’ll take them a tad more than 2.5 minutes to advance from 300 yards to 100 yards, where they’ll ostensibly charge the British.
The British infantrymen could probably fire at about 2-3 rounds a minute, so we’re looking at somewhere between 5 and 7 volleys before the Romans get into charging range. If the charge range is even less- let’s say 50 meters, it’ll take them another 38 seconds, and another volley or two from the British at murderous range before they charge.
Let’s say the Romans triple their speed in their charge by sprinting. That means they’re going roughly 10 mph, and can cover that 100 meters in 20 seconds. The British can potentially get ANOTHER volley in as they charge.
So in the best case, the Romans absorb maybe 5-6 volleys at increasingly short range before charging, and worst-case more like 9 volleys.
I’m not so sure that the vaunted Roman discipline would let them absorb that many volleys and the associated casualties without breaking. And even if they did, they’d probably be charging into a much greater number of Redcoats armed with bayonets.
As for the shield question, it’s been answered:
Range, Power, Penetration, Velocity of a Brown Bess – Roberts, Brown, Hammett and Kingston | Bow Vs. Musket (bowvsmusket.com)
Additionally they were able to penetrate 5" into an oak tree at 30 yds. To me, that translates into being able to penetrate a 1" shield at a much greater range.