BUG: None / Timedelta incorrectly returning NaT

doc/source/whatsnew/v1.1.0.rst

@@ -213,6 +213,7 @@ Timedelta
 ^^^^^^^^^
 
 - Bug in constructing a :class:`Timedelta` with a high precision integer that would round the :class:`Timedelta` components (:issue:`31354`)
+- Bug in dividing ``np.nan`` or ``None`` by :class:`Timedelta`` incorrectly returning ``NaT`` (:issue:`31869`)
 -
 
 Timezones

pandas/_libs/tslibs/timedeltas.pyx

@@ -1407,15 +1407,23 @@ class Timedelta(_Timedelta):
             # convert to Timedelta below
             pass
 
+        elif util.is_nan(other):
+            # i.e. np.nan or np.float64("NaN")
+            raise TypeError("Cannot divide float by Timedelta")
+
         elif hasattr(other, 'dtype'):
+            if other.dtype.kind == "O":
+                # GH#31869
+                return np.array([x / self for x in other])
             return other / self.to_timedelta64()
 
         elif not _validate_ops_compat(other):
             return NotImplemented
 
         other = Timedelta(other)
         if other is NaT:
-            return NaT
+            # In this context we treat NaT as timedelta-like
+            return np.nan
         return float(other.value) / self.value
 
     def __floordiv__(self, other):

pandas/tests/scalar/timedelta/test_arithmetic.py

@@ -412,6 +412,46 @@ def test_td_rdiv_timedeltalike_scalar(self):
 
         assert np.timedelta64(60, "h") / td == 0.25
 
+    def test_td_rdiv_na_scalar(self):
+        # GH#31869 None gets cast to NaT
+        td = Timedelta(10, unit="d")
+
+        result = NaT / td
+        assert np.isnan(result)
+
+        result = None / td
+        assert np.isnan(result)
+
+        result = np.timedelta64("NaT") / td
+        assert np.isnan(result)
+
+        with pytest.raises(TypeError, match="cannot use operands with types dtype"):
+            np.datetime64("NaT") / td
+
+        with pytest.raises(TypeError, match="Cannot divide float by Timedelta"):
+            np.nan / td
+
+    def test_td_rdiv_ndarray(self):
+        td = Timedelta(10, unit="d")
+
+        arr = np.array([td], dtype=object)
+        result = arr / td
+        expected = np.array([1], dtype=np.float64)
+        tm.assert_numpy_array_equal(result, expected)
+
+        arr = np.array([None])
+        result = arr / td
+        expected = np.array([np.nan])
+        tm.assert_numpy_array_equal(result, expected)
+
+        arr = np.array([np.nan], dtype=object)
+        with pytest.raises(TypeError, match="Cannot divide float by Timedelta"):
+            arr / td
+
+        arr = np.array([np.nan], dtype=np.float64)
+        with pytest.raises(TypeError, match="cannot use operands with types dtype"):
+            arr / td
+
     # ---------------------------------------------------------------
     # Timedelta.__floordiv__