feat(api): name each school's type group, and send search its nursery provision

The school page and the search rows now print a school's type in the search
filter's terms ("State school", "Independent school") instead of GIAS's 34
establishment types. The list, place and detail payloads carry type_group,
computed with the filter's own type_group_for, so the two never disagree; a
type in no group stays null and the page prints the register's name.

Search rows flag a nursery class, so nursery_provision joins SCHOOL_COLUMNS.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

backend/tests/test_type_group_payload.py

@@ -0,0 +1,81 @@
+"""Payloads name each school's type group.
+
+The school page and the search rows say "State school" or "Independent
+school" in the search filter's own terms, not GIAS's 34 establishment types,
+and an independent school gets a Fee-paying flag. A type in no group keeps a
+null group, and the page prints the register's own name for it. Search rows
+also carry nursery_provision, for their "Nursery class" flag.
+"""
+
+import numpy as np
+import pandas as pd
+import pytest
+from fastapi.testclient import TestClient
+
+
+def _schools_df() -> pd.DataFrame:
+    base = {
+        "local_authority": "Essex", "phase": "Primary", "year": 202425,
+        "ofsted_grade": 2.0, "ofsted_date": None, "attainment_8_score": np.nan,
+        "town": "Brentwood", "postcode": "CM13 1AA", "status": "Open",
+        "address": "1 Test Street", "latitude": 51.6, "longitude": 0.3,
+        "rwm_expected_pct": 60.0, "nursery_provision": "No Nursery Classes",
+    }
+    rows = [
+        {**base, "urn": 100001, "school_name": "Alpha Academy",
+         "school_type": "Academy converter", "nursery_provision": "Has Nursery Classes"},
+        {**base, "urn": 100002, "school_name": "Beta Prep",
+         "school_type": "Other independent school"},
+        {**base, "urn": 100003, "school_name": "Gamma Unit",
+         "school_type": "Secure units"},
+    ]
+    # Enough schools in one town for it to have a place page.
+    rows += [
+        {**base, "urn": 100010 + i, "school_name": f"Delta Primary {i}",
+         "school_type": "Community school"}
+        for i in range(3)
+    ]
+    return pd.DataFrame(rows)
+
+
+@pytest.fixture()
+def client(monkeypatch):
+    from backend import app as app_module
+
+    monkeypatch.setattr(app_module, "load_latest_school_data", _schools_df)
+    monkeypatch.setattr(app_module, "load_school_data", _schools_df)
+    monkeypatch.setattr(app_module, "get_supplementary_data", lambda db, urn: {})
+    monkeypatch.setattr(app_module, "_place_registry", None)
+    return TestClient(app_module.app, raise_server_exceptions=False)
+
+
+def _by_urn(schools: list[dict]) -> dict[int, dict]:
+    return {s["urn"]: s for s in schools}
+
+
+def test_the_list_names_each_type_group(client):
+    resp = client.get("/api/schools?page_size=50")
+    assert resp.status_code == 200, resp.text
+    schools = _by_urn(resp.json()["schools"])
+    assert schools[100001]["type_group"] == "state"
+    assert schools[100002]["type_group"] == "independent"
+    assert schools[100003]["type_group"] is None
+
+
+def test_the_list_carries_nursery_provision(client):
+    schools = _by_urn(client.get("/api/schools?page_size=50").json()["schools"])
+    assert schools[100001]["nursery_provision"] == "Has Nursery Classes"
+
+
+def test_the_school_page_names_its_type_group(client):
+    resp = client.get("/api/schools/100002")
+    assert resp.status_code == 200, resp.text
+    assert resp.json()["school_info"]["type_group"] == "independent"
+
+
+def test_a_place_page_names_each_type_group(client):
+    resp = client.get("/api/places/town/brentwood")
+    assert resp.status_code == 200, resp.text
+    schools = _by_urn(resp.json()["schools"])
+    assert schools[100001]["type_group"] == "state"
+    assert schools[100003]["type_group"] is None