As others have already said, you really do have limited options. If you are a quality stickler and retro-gamer enthusiast that really do want to get the best they can from the consoles they own and play, then your options are even more limited. Ideally with this many systems, you will probably be best off getting an external scalar such as the retrotink 5x (or 4k if you have the money, but really these are both for enthusiasts), or a XRGB-mini Framemeister (hmm... getting harder to find these as they discontinued them), or OSSC.
After that you are looking at modding the individual consoles themselves. Each of which will be costly to get a high quality output, but can sometimes be done on the cheap with something that will "work", but knowing it will not look as good as the original on TV's that had analog inputs.
The best quality way to deal with the Wii to get HDMI is with a fairly recent mod called "AVE-HDMI". This bypasses the internal digital to analog conversion that occurs and creates a direct digital to hdmi conversion, providing the cleanest looking outputs possible on the Wii. That said, it does not scale the output, as it is making a direct digital representation of the original output. It is recommended that you then scale this output with a hardware digital scalar to get 1080 and/or 4k output for modern TV's.
That being said, it is a lot easier to just pickup a Wii U which has HDMI output already and is backwards compatible with all Wii games (you do need to use the wii-mote and/or nunchuk, and need the sensor bar), but you will lose the ability to play Gamecube games on the Wii U vs on the Wii. Alternatively, if quality isn't that much of a factor, there are output adapters such as the Wii2HDMI, but the output quality is far inferior to the internal mod or simply using component and a quality scalar.
The easiest method for the PS2 is simply to get a PS3 60gb fat model that has hardware backwards compatibility that is in working order (and prepare to open it up, dust it out and re-apply thermal paste to the CPUs and GPUs). Now while this is the easiest way, it will also be a fairly expensive way as there is huge demand for these backwards compatible PS3's and there are also risks as they are prone to the "yellow ring of death" usually due to thermal damage (these can still go for almost $600 at times, but you can periodically get one for 200-400). This is the price you pay for having a single console that will still work with modern TV's and had full backwards compatibility to PSX, PS2, and PS3 catalogs....
As for the best output on the PS2, you will probably want to use RGB scart (there are several cables out there that will connect to the output) or component and connect to one of the aforementioned scalars.
The original XBOX is a little tricky since it's games used so many different resolutions. Probably the easiest is simply component cables with one of the scalars (although you may need to tweak settings on a per-game basis since as mentioned, as there are games that used everything from 480i, 480p, 576i, 720p, and 1080i). Alternatively there seems to be a HDMI mod that most people prefer (Stellar XboxHD+).
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Personally, I have an XRGB-mini Framemeister, and have modded most of my early consoles (NES, SNES, Sega Genesis) for RGB SCART output which is ideal for the XRGB-mini. For N64, I have the N64 UltraHDMI mod. My Dreamcast has the RGB VGA mod that I connect to the XRGB-mini. When I do use the PS2 (as opposed to the PS3), I connect via component to the XRGB-mini (I have been doing this more lately for PS2 and PSX games since the PS3 has been getting harder to obtain and I want to keep mine working, so putting less hours on it is a good thing). For my Gamecube, I connect via component cables (yes, I am one of the proud owners of the real component cables that were for the system, very few were made as almost no-one purchased them at the time, and now you will see that those original cables are worth more than the console itself, even now with a true alternative cables, the real ones still go for $200-450...).
The XRGB-mini had been the king of 480i and other interlaced content. While it isn't completely lag free like some of the other scalars (like the OSSC), it will de-interlace properly. Doing that usually saves lag overall (as the TV itself will then need to do the de-interlacing, and most of the internal processing on TV is not very good with interlaced video adding significantly more delay than what the XRGB-mini adds).
Today, if I had to do it all over again with today's prices on stuff, I would probably look at the retrotink 5x or the 4k. The stuff I have done and use was because I owned all those consoles initially and purchased most of those cables originally back in the day (especially the Gamecube). The RGB SCART mods were all very easy on the early consoles that I owned because most of those had intentions of possibly releasing with RGB SCART as an output, so I could easily mod the systems myself for just a few dollars in most cases. This made the XRGB-mini framemeister a logical choice once it released (and was only like $150-200).