The plan that SQL Server generated does not seem very good. It would have been better to merge-union the three tables and to the join to the IDs-table once. Maybe we can trick SQL Server into doing that:
SELECT t.cDT AS cDT, t.cID AS cID, t.c3 AS c3, t.cTAB as cTab
FROM
(
SELECT TOP 1000000000 *
FROM (
SELECT cDT AS cDT, cID AS cID, c3 AS c3, 'tab1' as cTAB FROM TABLE1
UNION ALL
SELECT cDT AS cDT, cID AS cID, c3 AS c3, 'tab2' as cTAB FROM TABLE2
UNION ALL
SELECT cDT AS cDT, cID AS cID, c3 AS c3, 'tab3' as cTAB FROM TABLE3
) x
ORDER BY cDT,cID --CI order
)
WHERE t.cID IN (SELECT ID FROM TABLEIDs)
ORDER BY t.cDT
This practically unlimited TOP clause might cause it to evaluate the union before doing the join. The order-by should help maintain the CI order of the base tables so that a sort operation is not necessary.
If this does not work right away, play with the idea a bit.