What would be the exact process?
Beware: I'm not certain this answer is correct (specifically whether an ouroboros sidechain could have $BTC as a UIA and the specifics of how the Bitcoin network would migrate across to using it).
Looking into this, Bitcoin could exist on Cardano as a sidechain with $BTC as a User Issued Asset (UIA). The nice thing about using Cardano sidechains is that transaction fees could still be in $BTC (instead of $ADA).
As to the exact process for achieving a migration, the answer lies with Ethereum's move to proof of stake and "the merge".
Image credit: consensys.net
The transition process would require the consent of client developers and exchanges in order to accept the changes. Consent of miners does not appear to be required.
However, Ethereum only exists due to Bitcoin's inability to evolve and make substantial changes. Unless something very significant changes in the Bitcoin world, it seems extremely unlikely that Bitcoin would attempt to chart a similar course. In addition, with so much money invest in equipment, mining would undoubtedly continue on a fork (similar to Ethereum Classic) raising the question of what would really be achieved.