Yeah, that actually should be fine because I'm sure PS2 is a component connection (red, yellow, white plugins). The PS1 is RF / Aerial, but even that works fine if your TV has it, which it should being that age. Its only the much older consoles that struggle on the RF connection.
I've realised not only did i buy the upscaler to HDMI, but I bought new wires that were component ones rather than be the one RF Aerial signal for my Megadrive and SNES.
This SNES one costs £5.99 and works actually for the Gamecube too.
Assecure replacement composite AV TV cable lead for Super Nintendo,Gamecube,64 (Super Famicom, Snes, GC, NGC, N64) : Amazon.co.uk: Electronics & Photo
WWW.AMAZON.CO.UK
Assecure replacement composite AV TV cable lead for Super Nintendo,Gamecube,64 (Super Famicom, Snes, GC, NGC, N64) : Amazon.co.uk: Electronics & Photo
This Megadrive one costs £6.99
CHILDMORY 1.8m 6ft Audio Video AV Wire Cable 9Pin to RCA Connection Cord for Genesis 2 3 Console : Amazon.co.uk: PC & Video Games
WWW.AMAZON.CO.UK
CHILDMORY 1.8m 6ft Audio Video AV Wire Cable 9Pin to RCA Connection Cord for Genesis 2 3 Console : Amazon.co.uk: PC & Video Games
So, really you can power your old consoles for next to nothing with an upscaler and a few newer wires.