(Hope it's not too late for a reply, because the test took some time)
Test-computer: Laptop with Intel Core i5-4200M CPU @ 2.50GHz / openSUSE Leap 15.0
Dolphin version from http://download.opensuse.org/repositories/Emulators/openSUSE_Leap_15.0/x86_64/
(dolphin-emu-5.0+git.1540367980.dc5a678fca-lp150.2.1.x86_64.rpm) 24-Oct-2018
All Dolphin-settings on default, Dolphin running with internal graphics. Although the Laptop has an Nvidia secondary GPU and optirun works for other applications, Dolphin did not work with it - so sadly no comparison what impact the better GPU would have had on the performance.
1.: GameCube
Super Mario Sunshine → full speed
Luigis Mansion → full speed
Mario Kart Double Dash → full speed
The GameCube games
were playable with graphics setting on 2x native (1280x1056) (sometimes slight drops in the frame rate). All other settings still on default. The CPU was not fully busy for GameCube games at native resolution.
Wii:
Super Mario Galaxy → full speed for the most time
New Super Mario Bros Wii → full speed
Super Smash Bros Brawl → full speed
The Wii games
were unplayable with graphics setting on 2x native (1280x1056) (except for New Super Mario Bros. This one is okay with 2x native.) Even at native resolution the CPU was at almost 100% for the most time but all three games were playable IMHO.
You should read the faq on the Dolphin-website. My laptop is five years old, but has a much higher clock speed than the computer you want to buy.
https://en.dolphin-emu.org/docs/faq/
This page states processors with “U” at the end are not good for Dolphin. They also say that newer generations are more efficient per clock. And they have a hardware forum for this kind of question. Maybe you can ask there?
If you do not need emulator specific features such as save states, it is a lot cheaper to buy a Wii and hack it.