N1P, I wonder why you would want to.
Flying the full ILS from 2000'/6nm down to the runway will take way more time than flying a visual circuit. Furthermore, since a lot of the airfields you will visit will have a relatively high proportion of VFR circuit traffic compared to IFR/ILS traffic, it will be easier for ATC to slot you in with the rest of the circuit traffic, compared to slotting you in on the ILS. And the last thing is that you need to remain VMC. So you have to fly around any clouds, which really is not an option on an ILS. So it's not exactly a practical approach method if you're VFR, unless there are no clouds whatsoever below 2500' and the traffic density at the field is such that you can be slotted into it via the ILS anyway. And in those circumstances, a "visual direct final" is probably easier anyway.
As far as training is concerned, yes, you can always request a practice ILS approach, or any other precision or non-precision approach for that matter. If you use the word "practice" ATC knows exactly what is intended. As you will still be VFR, it's up to you to remain VMC, and to keep a sharp lookout for other traffic as you will not be entitled to the same service as IFR traffic. So if you're going to be heads down during the approach, it's a very good idea to take a competent observer with you.
I fly from Rotterdam, a controlled field with a traffic intensity that I guess is about equal to Edinburgh. VFR traffic at our field usually flies one of the three visual approaches, but you can request essentially anything as long as you are willing to work with ATC and display an understanding of what will and will not be possible. When the runway in use is convenient, the weather is good, there is no IFR traffic approaching and I feel like it, I regularly request a "practice VOR/DME" or another IFR approach. I also regularly request alternate routings through the CTR, but I do make sure that my requests don't interfere with other activities in the CTR. So I make sure that I'm clear of the ILS areas that are in use, make sure I'm clear of the parachuting area, make sure that I'm not crossing a formal visual approach route at the designated altitude for that approach, and so forth.
So as far as practical flying is concerned, use a standard visual arrival or departure if they are anywhere near convenient. If not, request an alternate routing through the CTR but be prepared to work with ATC if for some reason your request is inconvenient or impossible. And if you want to practice a precision or non-precision approach make sure you use the word "practice" in your request, so ATC knows what the intention is.
BTW For formal IR training, with multiple approaches to the same runway, a lot of airfields require formal coordination, or even slots, arranged well before you get airborne. But in my experience a single practice request can usually be accommodated without formal approval in advance.