Slow Print Start

After the last couple of Octoprint updates, I've noticed that my printer seems to sits longer than it use to after bed leveling has completed and even after the bed and hot-end have achieved their set temperatures. Has anyone else noticed this?

Not noticed anything here (not that it should have changed anything) - pay attention to the terminal tab, to see when the printer sends back an ok response, to say it is done with the actions.

Seeing a lot of "Recv: echo:busy: processing", then finally Recv: ok

It's not that large or complex of a model that would take so much time to process I wouldn't think.

echo:busy processing is sent by your firmware. It's how it tells OctoPrint it's currently busy doing something, usually waiting for a heat up or bed leveling to complete. OctoPrint will and just not send stuff to your printer while it is signalling that it is busy. So if that busy period takes longer now, is probably been caused by either a firmware update our you might need a pid autotune to get your heat up back into whack. In any case you need to look at your printer and your firmware to solve this, OctoPrint is doing everything right here and listening to your printer.