Short answer (added on request): The differences between the standard languages of Germany and of Switzerland are minimal, mutual comprehension is very close to 100%, and misunderstandings are very rare. But dialect use is strong in Switzerland, 'Swiss German' is a collective term for the major Swiss dialects, and dialects from distant German speaking regions are not mutually intelligible in general.
When it comes to mutual intelligibility, it is crucial to distinguish the standard variety of a language and its dialects. This is because many varieties of a language are not mutually intelligible with each other.
Relation of the standard languages
German is often described as a pluricentric language, i.e. a language with more than one standard variety. The most important of these by far are (German) standard German (the form of German that people usually learn as a foreign language), Austrian standard German and Swiss standard German. In German, the standard forms of German are usually referred to as (deutsches / österreichisches / schweizerisches) Hochdeutsch. The three forms of standard German are mutually intelligible to about the same degree as standard varieties of English, i.e. almost completely.
I think the difference between standard German and standard Austrian German is roughly comparable to the difference between standard British English and standard American English, but with less differences of accent and of spelling. Even within German standard German there are often northern and southern expressions. The difference between southern standard German and Austrian standard German isn't much more pronounced than that between northern German and southern German. Nevertheless, nowadays it is more or less consensus in Austria that there is such a thing as standard Austrian German. On the other hand, I heard there are still school teachers in Austria who try to teach German standard German, and in doing so may prefer northern expressions that even southern Germans far from the Austrian border wouldn't use. Also, Austrian books tend to be published in German standard German; apparently the publishers want to target the northern German market as well and are a bit over-cautious.
The relation of Swiss standard German to German and Austrian standard German can perhaps best be described as similar to the relation of standard Indian English to standard American and British English. Mutual intelligibility is almost but not quite complete. The accents are quite different: Standard Swiss may sound more similar to Dutch than to German standard German. There are some lexical differences that can lead to difficulties, and there are very slight differences in grammar. Some of the differences can be observed in Swiss newspapers, but Swiss books are mostly printed in something very close to German standard German. Swiss orthography generally but not universally replaces ß by ss and word initial Ä, Ö, Ü by Ae, Oe, Ue. (This is because Swiss keyboards have to accommodate French accents in addition to the special German letters.)
Some of the lexical differences are a consequence of the fact that whereas Austria has shared a lot of history with Germany and was part of a centralised German Reich 1938-1945, Switzerland has an essentially unbroken tradition as an over 700-year-old federation of republics under French and Italian as well as German and Austrian cultural influence. The Swiss legal and administrative systems are quite different from those in Germany and Austria.
Relation of dialects
(Pure) dialects of German are not in general mutually intelligible with each other (or with the standard varieties), even within any of the three major German-speaking countries. Any English speaker who has been exposed to a pure local accent of British English such as Glaswegian will understand that this is true for British English as well. I guess it is also true for American and Australian English, but I haven't verified it. The purpose of the standard language(s) is to simplify communication between speakers of different dialects. Some dialect speakers have to learn the standard variety much like a foreign language, though not with the typical foreign language drills, but more informally along with reading and writing. Others pick it up much earlier from the media or are raised bilingually in dialect and standard variety. And in some regions the local dialect is quite close to the standard variety anyway. More importantly, in many regions of Germany the dialect has already died out or is in the process of being replaced by colloquial German with some regional influences.
In Switzerland the local dialects are not mutually intelligible with (even Swiss) standard German. They are also used much more extensively than in Germany or Austria. Apparently this started during the Second World War, when Swiss people had good reason to feel that it was embarrassing to speak German. It is worth repeating here that the Swiss dialects are not, in general, mutually intelligible with each other. When Swiss speakers from different regions meet, they have to adapt in some way -- probably by speaking a compromise between their own dialects and the standard language. Such compromises are also often used in the spoken media.
In practice, native German speakers from Switzerland have no difficulty at all communicating with Germans. In fact, the German that native speakers of the other national languages of Switzerland learn is generally standard German. However, Germans and Austrians, but particularly northern Germans, may have difficulty understanding some Swiss radio programmes and unless they are speakers of an Alemannic (south-west German or west Austrian) dialect they will be at a disadvantage in certain workplace situations as they will either be excluded or make others feel obliged to speak standard German.
Northern Germans in rural Switzerland
I am trying to answer the more precise updated question. I think the question itself severely understates the difficulty of understanding Glaswegian, or refers to standard English with a Glasgow accent and some dialect words as Glaswegian. How a local Swiss dialect probably feels to a northern German can perhaps be compared to how actual, proper, traditional Yorkshire English feels to a speaker of a standard variety of English: "Lowp over t'yat". As really pure dialects of English seem to be dying out, the relation of English and Afrikaans could be a better example.
