Consider this that it takes about 50-60 days for most peas/beans to produce. For example, if you plant some on july 1st, you will begin harvesting in late August. But beans are not like corn to come and go ALL at once. They keep fruiting for a long time. So, staggering peas/beans is not a big issue.
JMO

Thanks for your reply. My shell peas usually have a harvest period of about two weeks when I plant them all at once. I'm hoping to extend that by staggering them. I'm also planting varieties with different maturity days, so I'll are which works out better.

My shell peas usually have a harvest period of about two weeks when I plant them all at once

Don't they then reflower and produce more pods a couple weeks later, and on and on until the heat or white mold gets them?

I'm thinking of putting my peas in the bean trellises, then planting the beans 45 to 60 days later and letting them grow over the dead pea vines. Of course I'll clean off the pea vines if I can, but if the peas aren't dry by the time the beans cover them, it could be difficult to get the peas off without damaging the tender beans.

So I'm wondering if there is any chance of the white mold that usually infects dying peas getting transferred to the young beans?

Southerners have to deal with summer heat, which limits the time interval for English peas. I would not think that your summers are hot enough to limit you so staggered plantings should work. Planting varieties with different maturity dates is our best option and it does work. Two weeks is about the maximum harvest period for English peas. Those developed for machine harvest cut it a bit shorter.
This photo was taken a couple weeks ago. Willet Wonders planted Thanksgiving week, Premium planted late February. Both in bloom now.

Ditto on Farmerdill's comments. Beans grown for snap use will usually re-sprout & bear again, for several flushes if the season is long enough (and not too hot). Shell peas, on the other hand, are grown for mature seed. As seed matures, the plants enter their reproductive phase, and stop growing. While a few plants might re-sprout after harvest, the yield from these weakened plants would be negligible. Better to devote the space to something else.

It's worth mentioning, though, that there may not necessarily be a one-to-one relationship between planting dates & harvest dates. Seed planted in cool soil - like now - may take longer to germinate, and grow more slowly, than seed planted later. I call it the "catch up" effect; later plantings of the same variety tend to have a shorter DTM, and will gain ground on the earlier planting. There will still be a difference, but seed planted 2 weeks later might begin bearing only 1 week later.

My observation is that for peas, the earlier planting tends to be the healthiest. So as Farmerdill mentioned, it is best to stagger by using varieties with different DTM's (planted close to the same time) as opposed to using multiple plantings of the same variety.

Happy, I'm not sure I know the answer to your question re: powdery mildew spreading to beans co-planted. Never tried co-planting, or following one with the other. I have often planted peas & beans adjacently in the row, though, and never saw the mildew spread to the beans.