Walk to a bus stop. Seriously. Take a local bus to Hounslow or Hatton Cross. Then hop on the free airport shuttle bus or the Tube from there. It sounds insane, but it works.
Cheapest Ways to Get to Heathrow Airport - Swanky Travels
For example, bus #111 from Feltham goes straight to Terminal 4. A single bus fare is £1.75 with a contactless card. That’s two pounds. You cannot buy a coffee for that.
Yes, it takes forever. Yes, you might see parts of West London you didn’t know existed. But your wallet will sing hymns of joy.