Currently, it seems that the Elevenlabs integration waits until the entire speech file has been rendered before it plays. This can cause long delays when waiting for a response, as well as timeouts for very long responses.
The elevenlabs API supports streaming text to speech playback. It would be nice if the Home Assistant ElevenLabs integration used this instead of waiting for the entire response to be ready.
Don’t forget to vote for your own FR. 
This would be fantastic. Their v3 model is fantastic and would love to use it in HASS. But without streaming support it’s really too sluggish 
FYI,
Elevenlabs TTS now has streaming in 2025.11. It is not in the main release notes but mentioned in full change log: Full changelog for Home Assistant 2025.11 - Home Assistant
This uses both TTS streaming in Home Assistant in combination with streaming available from ElevenLabs to ensure playback starts really as soon as possible.