Low sensitivity is desirable because you want explosives to go off on demand, not randomly. Thus, most bulk explosives can be safely burned without detonation. Primary explosives have a low DDT threshold, meaning it takes far less energy and/or confinement to DDT. Thus, you can have a small primary, sometimes a secondary booster, to set off your bulk charge. The primer can be removed/dissolved to render the weapon far safer. But not perfectly safe.
I don't know the exact composition, but it is likely a combination of TNT, and one of the hexagen/MX explosives (RDX, HMX) [1]. Luftwaffe used Trialen 105 in their 250kg bomb [2], which is TNT, RDX, and aluminum powder, so lets go with that. TNT is fairly stable, *MX are quite stable, but picrates are ornery. Over time they can break down and react with metals in the casing. Metal picrates are much closer to primaries in temperament. They are also far more shock sensitive, which is why (somewhat paradoxically) it's safer to evac and burn these old bombs out, then to try to relocate it. There are other ways in which unstable residues can be formed, not just picrates (e.g. TNT can release NOx over time and form metal nitrate or fulminate salts). You also have all the impurities from synthesis (such as DNT and mixed nitrates). Over time this forms a witches' brew of mixed nitrates that's typically much more sensitive. Sensitive, as in, workers have died when their shovel struck the bomb casing. There would not even be time for the clunk sound to make it to their ears before obliteration. Or even just...nothing [4]. Hug your loved ones, folks.
(side note: this is why if you ever find old orndance, or old bottles of certain chemicals e.g. ether, you should stop immediately and don't try to move it, just evacuate and call a hazmat team)
What most likely happened is the old bomb corroded and formed a pocket of these unstable salts, and when the flame hit it, a DDT occurred which set off the bulk charge.
This is all well known to ordnance disposal folks, hence the sandbox cover and evacuation radius. [3]
Edit: tweaked wording after some more research into bomb composition.
P.S. I updated my notes on explosive composition. Turns out WW2 US/UK favored picrate mixes far more than the Germans (who favored TNT, RDX, ammonium nitrate, and aluminum), and as a result, allied bombs from this period are much more unstable than German ones. Also note, forming unstable species isn't necessary for DDT: e.g. the Halifax and Beirut explosions were caused solely by burning ammonium nitrate. It's just much, much harder to get a pile of AN to go off than rando metal nitrate salts.
[1] https://en.m.wikipedia.org/wiki/List_of_explosives_used_duri...
[2] https://en.m.wikipedia.org/wiki/SC250_bomb
[3] https://www.bbc.com/future/article/20150922-these-nazi-bombs...
[4] https://www.dw.com/en/wwii-bomb-self-detonates-in-german-fie...