The 38-gun frigate HMS Trincomalee (launched in 1817) and the 74-gun third rate ship of the line Implacable (launched in 1800 by the French as Duguay-Trouin, later captured by the Royal Navy at Traflagar) were collectively renamed Foudroyant and used as privately-owned training ships and accommodation ships from the late 1800s until after WWII. Implacable was scuttled after the war, but Trincomalee was saved.

The 74-gun third rate ship of the line HMS Wellesley (1815) was also being used as a privately-owned training ship. It was hit by German bombs on September 24, 1940 hit by one or more bombs during a German air attack near London. She caught fire and sank, making her the only man-o'-war ever sunk by an air raid! HMS Victory, although not in active service (she was undergoing renovations) was also bombed, although she was lucky to escape without serious harm.

On the "newer" end of things, there were the German pre-dreadnaughts Schlesien and Schleswig-Holstein (launched 1906), both of which were effectively destroyed at the end of WWII by Soviet mines or air attacks. There were two Norwegian coastal defense cruisers, Eidsvold and Norge, from 1901. Both were lost on the same day in April 1940 due to German attacks. The Japanese had the Tokiwa, which started life as an armored cruiser when launched in 1898, but was being used as a mienrlayer by the time she was bombed and wrecked in 1945. In a similar vein, there was the armored cruiser Izumoi which was launched in 1898 and was also sunk by American aircraft in 1945.

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Frankly it really does mean by what we consider being in active service, do we only care that it was in commission and that was used for something. DO we require it to have been underway, or is spending the war tied to the pier enough, do we require the ship to have been in active commission the whole time even if in secondary roles(and accepting temporary overhaul periods)?

As you note Constitution and Constellation were both placed back in commission and used for some duties. Both hosted change of command ceremonies (though Constellation was more often used for this) and Constitution was used at times as a brig for naval personnel in the Boston area. Ernest King for instance broke his flag on Constellation but soon was entrenched back in DC, where he had use of a new flagship, a refitted yacht that he could keep at the Navy yard allowing him to carry on at all hours and flirt rather shamelessly with other officers wives at times.

But frankly that isnt a particularly compelling service record and plenty of other ships of high vintage saw similar service. And HMS Victory of course even came under fire from the enemy, while others like HMS Warrior were is use as hulks as one sort or another. Others saw use as stripped down training vessels following massive reconstructions or refits. We can note the Russian cruiser Aurora as a good example in this category for instance.

But if the question is what was the oldest ship still in something like its original role that served actively at sea as a combatant its obviously a different question. We can look at the old IJN cruiser Izumo first laid down in 1898, and began the war still in a combatant role with a secondary formation off China, but even then was recalled to Japan in 1942 and took up training roles!

We can also look to Europe and there are a few ships that were in the fighting that predated 1900. Norway had a gaggle of coastal defense ships that tried to fight the German invasion for instance. Eidsvold was laid down in 1899 and attempted to put up a fight in the confusion of the first hours of the German landing but was sunk by a spread of torpedoes after she refused to surrender to a German destroyers. While the Germans also seized the former Dutch cruiser Gelderland that had been laid down in 1897, though again much of her role was for training cruises and instruction and only late in the war was she used as a mobile artillery and AA platform in Finland and sunk.

In the end depending on the criteria and how stringent we are there are probably a half dozen ships we could point to as the oldest to see service in WW2 depending on the specific definitions we use.