[core][scheduler] add missing process_to_add call (#20961)

Co-authored-by: J. Nick Koston <nick@home-assistant.io>

tests/components/core/test_scheduler_cancelled_callback.cpp

@@ -0,0 +1,44 @@
+#include "esphome/core/component.h"
+#include "esphome/core/hal.h"
+#include "esphome/core/scheduler.h"
+
+#include <gtest/gtest.h>
+
+#include <chrono>
+#include <thread>
+
+namespace esphome::core::testing {
+
+class SchedulerTestComponent : public Component {};
+
+// On a multi-threaded build next_schedule_in() only sees items_, so a timeout a callback leaves in
+// to_add_ would let a loop that sleeps on next_schedule_in() oversleep. The case that used to skip
+// the merge is a heap item (a zero timeout would run from the defer queue instead) whose callback
+// cancels its own item.
+TEST(Scheduler, TimeoutAddedByACancelledCallbackIsVisibleAfterCall) {
+  Scheduler scheduler;
+  SchedulerTestComponent component;
+  bool fired = false;
+
+  scheduler.set_timeout(&component, "first", 1, [&]() {
+    scheduler.cancel_timeout(&component, "first");
+    scheduler.set_timeout(&component, "second", 100, [&fired]() { fired = true; });
+  });
+  std::this_thread::sleep_for(std::chrono::milliseconds(5));
+  scheduler.call(millis());
+
+  auto next = scheduler.next_schedule_in(millis());
+  ASSERT_TRUE(next.has_value());
+  const uint32_t next_ms = next.has_value() ? *next : 0;
+  EXPECT_GT(next_ms, 50u);
+  EXPECT_LE(next_ms, 100u);
+  EXPECT_FALSE(fired);
+
+  std::this_thread::sleep_for(std::chrono::milliseconds(120));
+  scheduler.call(millis());
+  EXPECT_TRUE(fired);
+  EXPECT_FALSE(scheduler.next_schedule_in(millis()).has_value());
+  scheduler.trim_freelist();
+}
+
+}  // namespace esphome::core::testing