"Sorry, been busy" is true often enough to be believable and vague enough to be safe. What matters is what follows it: a busy person who is still interested proposes a next time. A busy person who is pulling away apologises and stops there.
The dynamic underneath: Busyness is the most socially acceptable form of withdrawal — it explains absence without admitting a change in feeling. It also protects the sender from a conversation they do not want to have. The reliable signal is not the excuse, it is whether they take responsibility for the next contact.
Replies you can send: Test for effort — "All good — when does your week calm down? I will hold a slot." | Low-pressure close — "Totally get it. Reach out when things ease up and we will sort something." | Ask for the honest version — "No pressure either way — I would just rather know if this has cooled off than guess."
How long should I wait before following up? Give it one message and a few days. If they do not propose a concrete next step after that, treat the pattern as the answer rather than sending another follow-up.
Is it rude to ask if they are losing interest? No, as long as it is calm and offers them an easy exit. Direct questions asked without blame usually get honest answers.