Job scheduling with a bottleneck problem
This looks like the so called master-slave scheduling model introduced by Sahni. In particular, your problem falls under the Single-Master Master-Slave Systems. You can distinguish several cases:

1) If you do not add any additional constraint on the order of job execution (as in your case), the problem is called unconstrained minimum finish time problem (UMFT) and has been shown to be NP-hard;

2) Same Pre- and Postprocessing Orders: it is possible to design an $O(n \log n)$ algorithm to construct an order preserving minimum finish time (OPMFT) schedule;

Therefore, in case 1 your problem is $NP$-hard, while it is in $P$ in case 2.

Additional related problems are:

3) Reverse Order Postprocessing: For any given preprocessing permutation, $σ$ , it is possible to construct a reverse-order schedule, referred to as canonical reverse order schedule (CROS). Given a preprocessing permutation $σ$ , the corresponding CROS is unique. It is easy to establish that every minimum finish-time reverse order (ROMFT) schedule is a CROS. 

4) no-wait-in-process constraint:

a) [MFTNW] Minimize finish time subject to the no-wait-in-process constraint;
b) [OP-MFTNW] This is the order-preserving version of MFTNW. That is, minimize finish time
subject to the no-wait-in-process and order-preserving constraints;
c) [RO-MFTNW] Minimize finish time subject to the no-wait-in-process and reverse-order
constraints.

Problems $a$ and $b$ are NP-hard, while $c$ admits a polynomial time solution.

Additional details in Handbook of Scheduling, chapter 17.