I am concerned that quantum computing could potentially brute-force all existing wallets. However, I am unsure whether internet connection speed would impose a limitation on such attacks. If not, how will the Bitcoin community protect itself against this threat? Additionally, how can I contribute, follow existing projects working on this issue, or learn more about this topic?
Quantum computing doesn't scale. It just generates noise.
Shor's algorithm, which is supposed to break ECDSA, fails miserably with any presence of noise,
even in theory. Hence the largest integer reliably factored up to now by a quantum computer using Shor's is 21. All those new and "very superior" QC managed only to generate more noise.