It sounds like this landing is complicated and my rule of thumb when things get complicated is to play is safe and glue everything. Especially on this landing which sounds like a very high traffic point, you'll want to reduce the potential for any squeaking, popping sounds and that's easier to do with a full spread glue down application (In my experience anyway). That's not to say you couldn't use screws or nails, but that introduces a fastener that may not hold or might start rubbing and making noise if there's a change in humidity and the landing expands or contracts. A small pail of polyurethane flooring glue (I use Mapei or Bona) and the recommended trowel would be more than enough - don't use PL as doesn't have enough play for expansion/contraction. Depending on material thickness at the nosings, you may even have enough room to add a thin layer of cork as an underlayment.
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