It's worth clarifying the difference between old-style cartridges, modern 3DS-style game cards, and memory cards (i.e. SD cards).
Cartridges - Used in NES, SNES, N64, etc - ROM - Directly addressable via the memory bus, meaning data doesn't need to be loaded into memory when starting up the game, giving close to zero load times. Impractical on a modern system for a variety of reasons.
Game Cards - Used in DS, 3DS, NX - ROM - Accessed via its own bus, assets must be loaded into memory before use. Modern cards should be comfortably faster than optical media, and potentially faster than HDDs for contiguous reads, and orders of magnitude faster than both in terms of latency. Not zero load times but potentially faster loading than systems with disc-based media.
Memory Cards - SD Cards, etc. - NAND Flash - Not used in games consoles. Slower than game cards due to the need to run everything through a flash controller chip.
NX will be using game cards, not cartridges or flash memory cards. The load times are going to depend on the interface Nintendo uses for the cards. The DS and 3DS use an 8-bit parallel interface, but this was chosen back in 2004 for a handheld with limited 3D capabilities, and had to accommodate writing saves to the card. If they're designing a new game card interface for a modern system it would make sense for them to switch to a serial interface and drop support for on-card saves. Hopefully they adopt a variable-speed serial interface, potentially allowing games that need it as much as 250MB/s or even 500MB/s (or higher for NX2) without increasing costs for games that don't.