A great production isn't limited to huge studios.
Also, no studio will turn down the opportunity for money. The general rule is, the better a reputation/equipment/engineer at a studio, the more you will pay. Of course, good shopping around will ensure you get the best bargain.
Record deals have nothing to do with getting a good song produced, unless they loan you money to do so, but it's not going to happen without some good tracks of your own.
There will be a lot of freelance engineers with an adequate setup willing to offer services at a low price (You must rememeber that a lot of outboard is unneccesary in the digital era, so you have to ensure that they have money spent in the right places. A good microphone (Worth £400 or more) is RECOMMENDED, however a skilled engineer can pull good results out of a cheaper one.
You may find that getting recordings for free, even from graduates may be difficult but not impossible. A person who is skilled enough to make money from it, may well end up making money out of it (Why not?). But you can expect to get a decent recording for around £50 if you look really well and ask the right producers. Soundcloud is a very good resource of VERY low budget music and how well they sound. Bedroom producers are a very easy target as they have their equipment for the love of making music, and would be happy with a small contribution, like myself!
Another trick you can manage, is if you can find a really amazing producer who's cheap to free, but they have a terrible microphone so you end up sounding rubbish, but you may even have a friend who has a decent mic, so get a good mic take, and get your allocated producer to make something with them.
I wouldn't trust Mihkays recommendation until I see something else other than price. The sound that they produce may well be amazing, but there is no indication as to whether they would have the right sound for you. They have many claims with no examples, it'd be nice to have a few audio demos on the site and maybe some tech specs.