I've verified the last test case locally; it took 8 minutes and 48 seconds.

How it works

çD Main link. Argument: n (integer)
 D Decimal; yield A, the array of base 10 digits of n.
ç Call the second helper link with arguments n and A.
Ḥçßç? Second helper link. Left argument: n. Right argument: A.
Ḥ Unhalve; yield 2n.
 ? If...
 ç the first helper link called with 2n and A returns a non-zero integer:
 ç Return that integer.
 Else:
 ß Recursively call the second helper link with arguments 2n and A.
ÆRDFṡL}i First helper link. Left argument: k. Right argument: A.
ÆR Prime range; yield the array of all primes up to k.
 DF Convert each prime to base 10 and flatten the resulting nested array.
 L} Yield l, the length of A.
 ṡ Split the flattened array into overlapping slices of length l.
 i Find the 1-based index of A in the result (0 if not found).

Alternate version, 11 bytes (non-competing)

ÆRVw3
ḤÇßÇ?

The w atom did not exist when this challenge was posted. Try it online!

How it works

ḤÇßÇ? Main link. Argument: n (integer)
Ḥ Unhalve; yield 2n.
 ? If...
 Ç the helper link called with argument 2n returns a non-zero integer:
 Ç Return that integer.
 Else:
 ß Recursively call the main link with argument 2n.
ÆRVw3 Helper link. Argument: k (integer)
ÆR Prime range; yield the array of all primes up to k.
 V Eval; concatenate all primes, forming a single integer.
 3 Yield the first command-line argument (original value of n).
 w Windowed index of; find the 1-based index of the digits of the result to
 the right in the digits of the result to the left (0 if not found).

Jelly, 17 bytes

ÆRDFṡL}i
‘çßç?
çD

Try it online! or verify the smaller test cases.

Jelly, 17 bytes

ÆRDFṡL}i
Ḥçßç?
çD

Returns the 1-based index. Try it online! or verify most test cases.