AFAIK the free eye tests are not conditional on you buying glasses, they are not allowed to do that. However, free eye tests are carried out by opticians not ophthalmologists and they are only allowed to test people with moderate corrections (eg: I have had laser surgery on retina and cornea and have VERY poor vision in one eye so regular "shopping mall" opticians are not allowed to test my eyes).
Specsavers have some good deals and their frames/lenses are as good as any other. Best thing to do is go in, get your eyes tested and ask for a quote. Generally they seem to speak English (at least in the HEV area they do). I have had several pairs of glasses from them (included varifocals, prescription sunglasses etc) and I'm very happy with them. Frames start at 9e (?) but it's the lenses that are more expensive. They offer different quality lenses at different prices, so you should be able to get the very basic ones at around 50-100e, maybe less depending on your correction.
I don't think your UK European Health Card will help you at all. But you might be able to get some student discounts....
Helsinki is full of shops offering discounts for eyeglasses. Try NISSEN shop, they will test your sight for free, providing you will buy glasses from them. Of course, this covers only "usual" glasses, ie. not for some special corrections needed. Ask for their offer, I got a pair of presciption sunglasses for 20 E on top.
Well Khan's in Oulu, but theres the same opticians. What I suggest for "cheap" is go get the eyetest/perscription, scan it and then go online to http://www.favoptic.fi (well, you need a Finnish friend as its in the local language only) - basically its like specsavers online, and about the same fugly eyewear. They send you up to three models, you try them on, send them and you get your glasses in the mail.
"By the pricking of my thumbs, Something wicked this way comes."