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JavaScript (ES7), 64 bytes

a=>(g=d=>n%--d?g(d):d<2?n**a.shift()*(a+a?g(++n):1):g(++n))(n=2)

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With BigInts, 65 bytes

A version that supports large integers. Expects a list of BigInts.

a=>(g=d=>n%--d?g(d):d<2?n**a.shift()*(a+a?g(++n):d):g(++n))(n=2n)

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Commented

a => ( // a[] = input array
 g = d => // g is a recursive function taking a divisor d
 n % --d ? // decrement d; if it's not a divisor of n:
 g(d) // do recursive calls until it is
 : // else:
 d < 2 ? // if d is less than 2 (n is prime):
 n ** a.shift() // multiply the final result by n to the power
 // of the next exponent extracted from a[]
 * ( // and multiply by:
 a + a ? // if a[] is not empty:
 g(++n) // the result of a recursive call with n + 1
 : // else:
 1 // just 1
 ) //
 : // else (n is composite):
 g(++n) // return the result of a recursive call with n + 1
)(n = 2) // initial call to g with d = n = 2