Quick answer: Early on, smash Broken Barrels and Trash Piles around your house with an axe — they drop Bronze Nails and Bronze Scraps. To make your own, unlock Bronze Works (Metallurgy) and build a Wooden Anvil: 1 Bronze Scrap becomes 6 Bronze Nails. Polished Bronze needs the Bronze Polishing tech and is made at the Woodworking Workbench: 2 Bronze Detail + 2 Sand + 1 Melted Fat = 2 Polished Bronze. You can also buy it for 75 bronze from the Furniture Maker, Jeweler or Smithy.
Players search this as polished bronze, polierte Bronze, polished bronce, bronze nails and how to create nails — they are the same bottleneck: most early buildings need nails, and the Writing Desk, Workbench II and Furnace II need Polished Bronze.
Bronze Nails
Loot barrels and trash piles
They sit in groups of two or three around your base, house and the village. Hit them with an axe for nails and scraps.
Unlock Bronze Works
First node of the Metallurgy branch.
Build a Wooden Anvil
1 Short Log (from the Sawhorse) + 1 Stone.
Forge nails
1 Bronze Scrap = 6 Bronze Nails. Later you can forge them from Bronze Ingots.
Polished Bronze
Unlock Bronze Polishing
It needs the Tool Rack upgrade on the workbench.
Make Bronze Detail
Forge it at the anvil. In the 100-day run the player needed Bronze Detail for the basement hatch as well.
Make Melted Fat
At the Distillation Cube: 1 Fat + 2 Clay Plates + 20 Fuel (Fuel).
Craft at the Woodworking Workbench
2 Bronze Detail + 2 Sand + 1 Melted Fat = 2 Polished Bronze.

Out of scrap?
Barrels and trash piles run out. For a steady metal supply see Copper; for copper-based items see the same page.
Related
- Writing Desk — needs 1 Polished Bronze
- Fuel — the Firewood Shed needs 2 Bronze Nails
Sources: Game Rant, wikily.gg, a 100-day playthrough on YouTube.