It is common but not universal for sites to accept, at a given time, 1) the current U2F token, 2) the U2F token from the previous window, 3) the U2F token for the next window. This is done as a partial mitigation for potential clock skew issues on the client that's generating the TOTP codes (e.g. your phone). In practice this means every code is valid for 1m30s, although sites may customize this (with or without changing the window size, which is typically not done because that parameter must be consistent system-wide).
> If I wait till the last few seconds before using the code, does that make me any safer?
Maybe, but this is not practicable security advice. The latency of a MITM attack on a 2-factor TOTP login depends on the attack infrastructure and design, but can easily be made to be on the order of tens or hundreds or milliseconds. Reducing the window seems like it might help your security, but it can never be perfect and there is a direct tradeoff with usability because users need time to look up the codes on one device and enter them on another.
Folks often say "enable 2FA" in response to news of new and sophisticated phishing attack campaigns, but it's critical to note that most commonly deployed 2FA (TOTP, HOTP, SMS) is trivially phish-able. 2FA is not an automatic defense against phishing, although some newer designs achieve this and were created specifically with this goal in mind: U2F is a good example.