This isn't related to cartridges. The kind of cartridge that avoided loading times was the one on pre-N64 consoles. It was mapped as part of the main memory and was connected to a fast bus. The modern cartridges are more like SD cards, so loading is necessary.
DF already benchmarked the Switch storage, the cartridges are slower than the internal flash memory or class 10 micro SDs: http://www.eurogamer.net/articles/digitalfoundry-2017-best-microsd-cards-for-switch-loading-time-comparison
They are faster than BRs, but PS4 stores the game in the HDD, so its storage is considerably faster than what is available on the Switch. So the difference is on optimization or the higher quality (and size) of the assets in the PS4 version.
Doesn't the WiiU version of BoTW have a mandatory HDD install?
I don't think PS4 and PS4 Pro's HDDs are faster than Switch's internal memory. Switch's internal has a max of 400MB/s. Upgraded models can have 1200MB/s. It's also possible that publishers choose both the size and speed of the cartridges depending on need.
Switch's internal memory (PDF download, part:THGBMHG8C2LBAIL): https://toshiba.semicon-storage.com/info/docget.jsp?did=12587
Don't Play Stationary 4 ever. Switch!