[api] Increase noise handshake timeout to 60s for slow WiFi environments (#15022)

esphome/components/api/api_connection.cpp

@@ -64,7 +64,11 @@ static constexpr uint32_t KEEPALIVE_DISCONNECT_TIMEOUT = (KEEPALIVE_TIMEOUT_MS *
 // A stalled handshake from a buggy client or network glitch holds a connection
 // slot, which can prevent legitimate clients from reconnecting. Also hardens
 // against the less likely case of intentional connection slot exhaustion.
-static constexpr uint32_t HANDSHAKE_TIMEOUT_MS = 15000;
+//
+// 60s is intentionally high: on ESP8266 with power_save_mode: LIGHT and weak
+// WiFi (-70 dBm+), TCP retransmissions push real-world handshake times to
+// 28-30s. See https://github.com/esphome/esphome/issues/14999
+static constexpr uint32_t HANDSHAKE_TIMEOUT_MS = 60000;
 
 static constexpr auto ESPHOME_VERSION_REF = StringRef::from_lit(ESPHOME_VERSION);