India and Pakistan.
India most likely completed weaponized nuclear warheads around 1994 and tested them in 1998. Pakistan also tested their first nuclear weapons in 1998.
Both had a war in 1999, the Kargil War that lasted almost three months and resulted in a few hundred (official figures) to a 1000-2000 (claims) casualties on each side.
So, for your question they both would count unless you think that is not significant enough.
As another potential case: Israel is widely believed to have nuclear weapons since ~1960-63 and was since then involved in the Six-Day War, the War of Attrition
, the Yom Kippur War, Lebanon Wars, Intifadas, Gaza Wars and some smaller stuff like being bombarded by missiles from Iraq. For a list of all involved countries see this table.
So if you believe that Israel really has nuclear weapons and is a nuclear-"protected" country, then Israel would definitely be such a country.