Nokia Lumia 510 is a good smartphone so there is a very little chance that modules like Bluetooth, WiFi or FM will malfunction unless something terrible has happened to the phone like if it’s dropped or water damaged.
Before making any hardware repairings reset your phone just to make sure that some user app or setting is not causing the problem. Here is a guide on How to hard reset Nokia Lumia 510.
There is not a lot to explain for this solution I have marked all WiFi, Bluetooth and FM modules in the diagram below. Re hot those modules carefully, replace if nothing else works.
Here are Nokia Lumia 510 Bluetooth, WiFi and FM modules marked out. Please study these diagrams very carefully before you start repairing.
If you have further questions please don’t hesitate to leave a comment.