Would uncompleted post secondary education affect the chances of getting postgraduate studentships? For example, a student who has switched university course after one year.

If you then went on to achieve a good honours degree (i.e. 1st or 2:1) your chances are good. I wouldn't even mention it, plenty of people complete a degree a year later than other people the same age, due to having spent a year out travelling or A-level re-sits etc.

It shouldn't matter, as long as you completed the degree second time around.

nope i dropped out of a pgce and had no problem getting my phd place.

Does it need to be declared on a CV, if not asked specifically? and would it suggest a lack of preserverance??

I wouldn't specifically refer to the abandoned course on your CV. If you are asked about it (why you spent 4 years doing a 3 year course or similar) you can then explain. No point in pursuing something that you decided was not right for you or your future career plans. The successful completion of your subsequent course demonstrates that you have the commitment required to see something through.

Is it legally required to include that detail in the CV? Or can it be ommited if not asked about?

You can, of course, get into big trouble for lying on a CV, but no one is advising that you do that! What you put on your CV is up to you, and in the pursuit of brevity, we are advised to include only the information that is relevant to the application. IMHO, the fact that you once withdrew from a course has little relevance to your application. I would devote the precious space on your CV to the important things, like the degree course that you did complete.

Of course, if you are asked directly, you will have to own up but I think that that is very unlikely to happen (no one has ever asked me 'have you ever started a course but not completed it?' at any job or PhD interview and I've certainly never seen that on an application form). Don't worry too much about it, good luck with your application :o)