I only have a BIOS, so I don't think that my Windows is installed in UEFI. Also, when first tried installing Ubuntu, I chose the UEFI way
This statement is internally inconsistent. If you could choose a UEFI boot option, then you have an EFI, not a BIOS. (Note that UEFI is Unified EFI, aka EFI 2.x. I use "EFI" to refer to any variety of EFI, including UEFI, as "EFI" is the more general term.) You're probably confused because of the bad habit of many manufacturers to refer to their EFIs as BIOSes. I suspect they do this because many people have at least a vague idea of what a BIOS is, and as an EFI fills a similar role, it's easier for the manufacturer to use the term "BIOS" where they really should be using "EFI" or "UEFI"; but this practice creates confusion because people like you see "BIOS" in the documentation or computer's screens and think they've got a BIOS. People also see "BIOS" and drag in BIOS-specific knowledge that's just plain wrong on EFI-based computers. In fact, though, BIOS and EFI are two entirely different types of firmware, much like a pickup truck and airplane are two entirely different types of vehicle.
You may want to read more background information on what EFIs are, how they work, and how they differ from BIOSes:
I chose the UEFI way and it told me, that windows was not installed in UEFI and that there will be a problem, so I chose the normal (SATA) way and then it didn't show any option.
This reveals more confusion over terminology. Serial ATA (SATA) is a hardware standard for connecting hard disks to computer motherboards or disk controller cards. It makes no sense in the context in which you've used it. I suspect you mean BIOS/CSM/legacy.
It's certainly possible to install Windows in BIOS mode, but all computers that shipped from major manufacturers (Dell, HP, Acer, Sony, etc.) with Windows 10 installed shipped with it installed to boot in EFI mode. Thus, chances are you either installed Windows yourself in BIOS mode or you're misinterpreting or mis-remembering the error message you saw. It's hard to know which of these is the case (or if it's something more exotic) without more information, and understanding your current configuration is critical for fixing it.
I realize that this is a long response and it doesn't come close to answering your question. Unfortunately, you haven't provided the details necessary for me to do more than guess about your real issue. If you want more help, please run the Boot Repair utility from an Ubuntu installer in its "try before installing" mode and select the "Create BootInfo Summary" option. (DO NOT click "Recommended Repair," at least not yet!) When asked whether to upload the report, click "Yes," and then post the URL provided here. This will give us more details about your configuration, which is required to base an answer on more than guesswork.
EDIT: Based on your Boot Repair summary, it looks like your computer is EFI-based, but both Windows and Ubuntu are installed in BIOS/CSM/legacy mode. GRUB (the Ubuntu boot loader) is installed to the MBR of /dev/sda, and the Windows boot loader is installed to the MBR of /dev/sdb. Ubuntu and Windows seem to be installed mostly to /dev/sda and /dev/sdb, respectively, although there's one NTFS partition on /dev/sda.
Thus, it looks like your system is set up correctly, at least in terms of on-disk data structures. Chances are your booting straight to Windows is caused by the boot order in your firmware. I recommend launching the firmware setup utility and adjusting the boot order so that what Linux calls /dev/sda (a Hitachi HDP72505 disk) is first in the boot order. Unfortunately, the details of how to do this vary greatly from one computer to another. You usually access the setup utility by hitting Esc, Enter, or a function key shortly after powering on the computer, before the GRUB menu or any OS-specific messages appear. If you need more help with this, or can't find a boot order option, you should ask the manufacturer or ask on a manufacturer-centric forum.
As you asked about it, it is possible to convert both Windows and Ubuntu to boot in EFI mode; however, the process is tedious and runs a risk of making matters worse. Thus, I don't recommend doing this unless you become desperate -- and even then, only if you back up your important data first. You could try this new Microsoft tool or this older procedure to convert Windows from BIOS-mode to EFI-mode booting. For Ubuntu, you could install my rEFInd boot manager after converting Windows or run Boot Repair from an EFI-mode boot (which is how you booted it to get the summary data). In either case, you'll need an EFI System Partition (ESP), which the Windows conversion process should create.
