It shouldn't cause any huge problems for you beyond longer loading times. It's recommended, not required.

For using external storage, you could in theory run the game off a usb stick if it had enough storage, or an external hard drive, which plugs into usb as well.

Good luck!

quotes from twitter:

CDPR: It’s mainly streaming issues - when the settings are too high for your setup the game is unable to stream the content and stuff starts to go missing (animations, audio, lipsync, objects). Adjust settings and it works waaaay smoother.

twitter user: Hey would a fast ssd help with this problem?

CDPR: Yes, absolutely. You will run in another bottleneck at some point though if you set everything to max / ultra. It seems like many people don‘t understand that this game can make every (!) PC struggle if you go crazy with the settings. Not many games have done this lately.

looks like you need to go with lower settings when you're on an HDD

Source: https://twitter.com/fabiandoehla/status/1336116453867728896

As poster above said.. higher setting require a larger bandwidth, if running on HDD vs SSD vs nvme you'll see significant increases in speed as you go up the list... HDD might do 6Gb/s on sata, but the 'seek times' and drive latency CAN (not always depending on disc placement of resources) slow you down some, if you also have lower memory, then not as much can be cached. 6Gb/s SSD is faster not because of actual max bandwidth, but the drive technology is MUCH faster on latency and disc seeks, and nvme m.2 is even faster as it's 32Gb/s with all advantages of sata SSD.

A millisecond seek time disadvantage doesn't seem bad until you have to do 10,000 of them, then it's 10 seconds and so on.

Assuming you mean USB attached external storage, this can lead to MANY other problems, there's a reason the current generation of consoles don't even allow this now. USB can 'just disconnect and reconnect' sometimes, and drive latency is horrible in comparison to internally attached drives. If you have a pci externally attached drive it's not subject to same issues, but usb 2 is limited to much lower bandwidth than sata-III hdd or ssds and the latency can be an order of magnitude higher. If you MUST run off usb attached storage, expect issues as code will generally have some 'performance' expectations that are not likely to be met by USB storage. In my systems, USB external drives are for things like images, or backups and not apps or games that are being actively used.

That's not quite true. No HDD actually achieves 6Gbps (=750MB/s theoretical limit, reality is more like 550 to 600), good HDDs top out at about 150 to 200 MB/s sequential read/write. SATA SSDs can reach these speed of ~550 MB/s though.
NVMe SSDs are faster, because they use the 32 Gbps (= 4 GB/s theoretical limit) interface, but also because NVMe is a new protocol that was specificly designed for SSDs, while the SATA protocol is from a time where SSDs basically didn't exist yet.

latency and seek times are many times faster on SSDs than on HDDs like you said. This increase is mainly the reason why the operating system and application run so much snappier with SSDs. NVMe SSDs are also fast in this than SATA SSDs, but in most cases it's barely noticeable, because the difference is only fractions of a second.

:)

I know, I'm a systems architect (with some storage background) but I can't unload ALL that info.