When playing an N64 game on Wii virtual console, what is happening when the spinning N logo appears?

It's loading both the emulator and the entire N64 ROM into RAM, and also loading a firmware handler to run the console in "N64 mode." Most emulators on Nintendo's systems (beyond the simpler NES and SNES emulation) have a special firmware like this, and switching between that and the normal Wii Menu firmware would account for the delay when using the Home button.

Another example of special firmware is the 3DS firmware for running GBA games.

15 seconds seems excessive, and I remember it being a bit shorter than that on Wii U N64 VC games, but I would imagine that N64 emulation on Wii was rough around the edges, being an early step in emulation for Nintendo (although Nintendo staff weren't the ones who developed it).

N64 games are not particularly large or memory-intensive compared to native Wii titles

That's true, but note that original N64 games can be read directly from the cartridge with speeds comparable to RAM. The Wii stores the N64 ROMs on its internal flash memory or an SD card; both kinds of memory are faster than, say, discs, but not as fast as cartridge memory was. So the N64 has a speed advantage in not needing to really load anything, but just boot up and jump right to the title screen.

...loading from disc (which doesn't have this behavior either)

With a Wii game, time to boot is faster because there is no extra firmware and not much necessary to load into RAM initially, but after boot, many Wii games have protracted opening logos or loading spinners (or both!) that mask the wait time as content is read from the disc.