FrozenNorth
Hero
As someone who actually enforced this (not at the scale of cities, but at the scale of municipal buildings and palaces), natural weapons are pretty useless even in this case.I was talking about visitors to the city. While they might very well allow an old man his staff, they're not necessarily going to allow a group of well armed mercenaries to do the same. A craftsman might be allowed a hammer but a fighter might not. It makes sense that in a world where even a dagger can be a powerful and dangerous magical weapon, that guards might be wary of allowing dangerous looking strangers even that much.
Let’s put aside what I expected the players to do (which they didn’t): put their weapons in Leomund’s secret chest for retrieval in cast of emergency.
Barbarian: impacted, unless grabbed a table leg for an improvised weapon;
Bard: relied on cantrips
Cleric: relied on cantrips
Druid: Wildshape, and relied on cantrips;
Fighter: impacted except for the Eldritch Knight, who summoned his weapon through Eldritch Bond (or grabbed a table leg for a weapon);
Monk: they’re happy that everyone is in the same boat as they are.

Paladin: Smite doesn’t care that you are punching for 1 pt damage;
Rogue: used Sleight of hand to hide dagger
Ranger: impacted (unless used Sleight of hand to hide dagger)
Sorcerer: relied on cantrips
Warlock: relied on Eldritch blast, except for the bladelock, who simply summoned his weapon to his hand;
Wizard: cantrips.
Out of 12 classes, 3 were impacted, and even they had really easy ways to mitigate the effect of not bringing weapons with them.