It's not clear what you are proposing. Are these speakers built into the TV? If so, the TV will have to be switched on, so you might as well use its phono inputs, if it has them.
If these are separate speakers, are they active or passive? Are they connected to an amplifier? Does that have phono inputs?
You seem to be suggesting using analog switches. This can only be done with line-level signals. If the signals are amplified, it could be done with relays.
Most types of Arduino cannot deal with bluetooth audio data. They are not fast enough.