Issue #19946: use public API for multiprocessing start methods

This should appease the OpenIndiana buildbot.

Also lengthened the worst case timeout to try to eliminate
the inconsistent failure on one of the Windows 7 buildbots.

Lib/test/test_multiprocessing_main_handling.py

@@ -15,9 +15,9 @@ from test.script_helper import (
     assert_python_ok, assert_python_failure, temp_dir,
     spawn_python, kill_python)
 
-# We look inside the context module to find out which
-# start methods we can check
-from multiprocessing.context import _concrete_contexts
+# Look up which start methods are available to test
+import multiprocessing
+AVAILABLE_START_METHODS = set(multiprocessing.get_all_start_methods())
 
 verbose = support.verbose
 
@@ -51,7 +51,7 @@ if __name__ == '__main__':
     p = Pool(5)
     results = []
     p.map_async(f, [1, 2, 3], callback=results.extend)
-    deadline = time.time() + 2 # up to 2 s to report the results
+    deadline = time.time() + 5 # up to 5 s to report the results
     while not results:
         time.sleep(0.05)
         if time.time() > deadline:
@@ -79,7 +79,7 @@ set_start_method(start_method)
 p = Pool(5)
 results = []
 p.map_async(int, [1, 4, 9], callback=results.extend)
-deadline = time.time() + 2 # up to 2 s to report the results
+deadline = time.time() + 5 # up to 5 s to report the results
 while not results:
     time.sleep(0.05)
     if time.time() > deadline:
@@ -131,7 +131,7 @@ class MultiProcessingCmdLineMixin():
     maxDiff = None # Show full tracebacks on subprocess failure
 
     def setUp(self):
-        if self.start_method not in _concrete_contexts:
+        if self.start_method not in AVAILABLE_START_METHODS:
             self.skipTest("%r start method not available" % self.start_method)
 
     def _check_output(self, script_name, exit_code, out, err):