Are LED bulbs for brake lights & indicators road legal? Do they need extra resistors?
Yes and No.
All LEDs require a series resistor to limit the running current. This is usually built in and isn't what is normally meant when describing an automotive LED package as having a 'resistor'.
LED replacements for indicators will almost always need to have power-wasting parallel resistors to take enough current to fool the flasher unit that a regular bulb is in use, else it won't flash properly.
LED replacements for other bulbs may also need parallel resistors to dump enough current to fool any bulb-fail detection circuits that all is well.
Legal situation is unclear. The UK laws don't mention LEDs at all, but specify bulbs of whatever wattage. So in theory retrofitting a car with LED mandatory lights is illegal. Many of course do pass MoTs with these units fitted. A consideration though is that LED units tend to have 'beam' type pattern ; when used with lenses designed for real bulbs the viewing angles will be wrong. A sharp-eyed traffic cop or MoT tester could just pick this up.
New cars are type-approved as a whole item, including any LED lamps factory fitted, so they don't have to comply with the older laws.
