Must this adversary (or any data collector) be watching me live, exactly to the time point, if I would make the swap with ETH?
No, not at all. Given that both Bitcoin's and Ethereum's blockchains are completely public, anyone can look back at any point in the future and try to link your transactions together.
Given, there was no data collecting at the time I made the switch from BTC into ETH and back, then the only possibility to reconstruct the trace would be