While you don't have to rush right out and move the pile immediately, I wouldn't recommend leaving them in place indefinitely. 4-5" is a bit thick for mulch over tree root systems - I'd not go more than 3". Applying them directly as mulch now is of little concern - it's only when they are worked into the soil that any issues of nitrogen depletion arise and that can be easily compensated for. They also make a great path material through planting beds or other, informal areas and keep mud to a minimum.

Fresh material generates white smoke-like clouds when disturbed that are probably a good potential source of aerosol pneumonitis (farmer's lung). These appear soon after delivery. This might be its own motivation for getting it all spread promptly - unless most or part of it had already been molding before you got it.

If not ground within the last few days or so, consider wearing a mask when spreading.

The material was ground on the day it was delivered (3/2). But it may be a few weeks before I can get to moving it. Does the same apply to garden bark?

Yeah, that molds also. But wood chips really get smokin'.

It's the fresh foliage that may be mixed in with wood chips that really kick it out. If you got a cleaner product, like cedar 'play chips' from a mill you won't see it.