How much memory has the system got?
Are you shutting it down or putting it into hibernate/sleep? I believe Windows 10 defaults to a sort of "quick boot" state and loads most of it's memory from disk at start up.
When using task manager to diagnose, you need to open the advanced details view and look for the Disk column. Click it to sort by disk usage. If nothing appears to be using the disk, check the performance tab and look at the disk graph. If the HD is trashing and nothing appears to be claiming it, it's most likely the windows kernel moving memory pages around.
I'll take the flack for this, but Windows memory management is absolutely terrible, especially when running low on it. It tries to be efficient by caching and recaching things in the page file, but that causes huge bottle necks when you "run out" of memory and have things being offloaded TO disk in order to reload cache or files OFF disk. Forcing the HD to carry out 2 operations at different locations on the disc causing "trashing" which is extremely slow. The whole thing generally causes it to build up getting slower and slower and then only thing you can do is go away and let it finally catch up.
The minimum for a Windows 10 PC these days is 8Gb of RAM. If you have 4Gb you will experience issues running any sizable professional application, like LightRoom, Photoshop or video editor. 16Gb and most memory issues go away unless you are editing HD or 4K video and need a massive in memory disk cache for scrubbing.
I was running Windows 7 on my laptop, i3 6Gb RAM. It did exactly what you are describing, 100% disk for 5-10 minutes after coming out of hibernate. Really sluggish performance, even to open Chrome and YouTube. I used to turn it on, before getting ready for bed, so it was ready for late night viewing 5 minutes later.
I upgraded the laptop to an SSD, nothing special, just a £30 Corsair 256Gb. I installed Windows 10, which turned out to be pre-licensed on that laptop.
Night and day. My SSD Windows 10 systems ALL take longer to get through BIOS to the windows boot logo than it takes Windows to boot to desktop. I can hit the power button on the media centre, then the TV, then by the time I have jumped into bed and fixed the pillows, not only has the Windows desktop appeared, but it's nicely relaunched Chrome at the last tab as well.... in under 15 seconds.
If you can't go for the upgrade to SSD and/or more memory, a completely wipe and reinstall of Windows 10 would also be a temporary option. It will clean out all teh redundant crud that builds up in the start process and registries.