Quantum computers will pose a threat to ECSDA. Whilst 256-bit ECC is comparable in strength to 4096-bit RSA, to a quantum computer, all it's concerned with is the length of the numbers. A 256-bit integer is far easier to solve for compared to 3072-bit.. A single signature would be enough to compromise a private key, with a strong enough quantum computer. They pose less of an issue to hash functions, so funds received on a bitcoin address are safer (until the first transaction redeeming them reveals it's public key and signature)
While quantum computers of this size aren't practical right now, they soon will be. I read a journal article documenting a quantum computer for factorizing integers using chemical computers. Nuclear magnetic resonance was used to induce quantum states in a molecule containing 5 fluorine atoms (used to store qubits of information). It's a pretty extreme approach (they won't break ECDSA using this setup), but it was also largely successful. (
https://cryptome.org/shor-nature.pdf)
We're mainly waiting on something that helps us realize quantum computing to a practical extent, but upgrading bitcoin to use a new signature algorithm can be accomplished by either a soft or hard-fork if preferred. With all systems, they will be upgraded whenever the risk becomes real.
wrt upgrading, we can only make it opt-in, so softfork is probably best. Anyone who has funds protected by ECDSA would move their coins one, to a new scriptPubKey protected by: OP_PQCHECKSIG (post-quantum checksig, whatever we decide to adopt). We would generate a new address type, starting some other prefix besides 1...., or 3....., and then life would carry on!