You are confusing the terminology a bit.
UEFI and BIOS are not partitioning schemes but refer to the style of firmware on your motherboard - specifically, UEFI is a replacement of BIOS, and brings new UEFI boot mode, but UEFI can still boot in the old BIOS boot mode too.
The distinction in partitioning schemes is DOS/MBR style partitions vs newer GPT style partitions. GPT is a newer scheme that is not limited to 4 primary/extended partitions.
To boot in UEFI mode, you need to be using the GPT partition scheme and have an EFI partition. However, if you use GPT partitions you do not need to boot in UEFI mode. How you boot is a function of how the operating system has installed its boot manager. Modern versions of Windows will automatically realise that you are using GPT and that the system is capable of UEFI boot and install a UEFI boot manager. Depending on the version, Ubuntu may do this too.
There is nothing stopping you from repartitioning using DOS/MBR partitions, and re-installing Windows and Ubuntu. It would force everything to boot in BIOS compatibility mode. Note that you would need to reinstall Windows from its original installation disc, as the boot type is determined at installation.
You don't, however, need to do this to get dual-booting working between Windows and Ubuntu. It's perfectly possible to dual-boot between Windows and Ubuntu, both using UEFI boot. Many other questions on this site deal with this. It can be difficult sorting out the problems, but there are no more problems than when we used to boot using MBR - in both cases Windows setup/repair liked to overwrite the existing boot manager with its own, making it necessary to re-install Grub's boot manager after running Windows setup. In this case you just do it with EFI rather than an MBR boot manager.
If you do want to get UEFI working please ask a new question and give more information about your system and setup, but if you don't - the short answer is that you are welcome to repartition using DOS/MBR.
/Microsoft/Windows/BOOT/
and rename them tobootmgfw.efi
andbootmgr.efi
Themgfw
one is the 64-bit file. Rename the Windows files to something else and runsudo update-grub
to see if they're found. – TheWanderer Sep 10 '15 at 00:01