What is a polite way to tell a Scottish person that I am struggling to understand his/her accent?

From personal experience I can say that asking politely to repeat what they just said helps the most. “Beg your pardon?”, or “I’m sorry, what?” or “I’m terribly sorry, but I didn’t understand you” all worked very well for me. I never had the feeling that I offended the speaker.

I’m English and lived in Glasgow for 4 years. Understanding many (but certainly not all) people will be tricky (even us native speakers struggle), but they’ll be very willing to try to help you understand, and it won’t cause offence.

You probably won’t even need to explain: as soon as they see look of blank incomprehension and hear you begin to say “Err, sorry” in a non-Scottish accent they’ll usually laugh and know what’s going on. It’s a common situation. A few things to understand about Glasgow:

• They’re used to it. It’s a running joke in the UK that the Weegie accent is especially hard to understand. I certainly struggled more than once, and I’m a native English speaker. Everyone in Glasgow is familiar with this. Heck, plenty of people from Edinburgh struggle to understand Glaswegians (to the delight of many Glaswegians, who take it as proof that their Edinburgh rivals are less truly Scottish…). It’s a very familiar situation.
• It’s actually somewhat exaggerated – most Glaswegians are very easy to understand. So much so that Scotland generally and Glasgow in particular is a popular place for companies to house call centres – it’s a very expressive accent, and studies find people respond well to it, it sounds honest and warm. But when an individual Glaswegian is difficult to understand, and many are, they’re very difficult to understand – and you will encounter several such people. It won’t be everyone you meet, or even the majority, just a very memorable, sizeable minority.
• They take pride in being helpful to foreign guests. Glaswegians take a lot of pride in being welcoming to outsiders (especially non-English outsiders… :-) but surprisingly welcoming to us sassenachs too), and particularly to Scandinavians (many pro-independence Glaswegians feel more affinity to social-democratic Scandinavia than they do to the London-dominated, more conservative UK). The Refuweegee project is a topical textbook example of Weegie pride in being warmly welcoming to foreigners. Most Glaswegians will be very happy to try to help you out.
• They’re very aware of the difference between their speech and standard English, and tend to be rather proud of it. The idea that Scots dialect is not a dialect but a wholly separate language to English is pretty popular in pro-independence Glasgow, and there’s even a term for and books about Glasgow’s unique “patter”. Scots spell differently when writing in Scots as opposed to formal English: Scottish Twitter is great, and there are even English-Scots online translators. Glaswegians take a lot of pride in having a good sense of humour and being able to take some stick, and also in being a bit different to everywhere else. Something like this won’t cause offence.
• They may have had the same experience themselves. Not only are Glaswegians familiar with others struggling to understand their accents, they’re also familiar with struggling to understand other, even stronger Scottish accents themselves. For example, there’s the potentially-offensive Scots word Teuchter (pronounced like Choo-chter, with the second ch like Loch), which a Glaswegian friend cheerfully translated to me as meaning “Those bampots from up North who we cannae understand what they’re saying”; before doing an impression of a Teuchter accent (north highland Scottish) which, to me, was actually slightly easier to understand than her normal speech… Scottish accents are very varied, and it’s not uncommon for Scots to have experienced struggling to understand other Scots.

While getting people to try to help you understand will be easy, getting them to succeed in helping you understand them is a different story… Luckily, if all else fails, Glaswegians tend to be very expressive with their body language :-).

You’re very unlikely to cause offence (unless you accidentally get your country names mixed up and talk as if Scotland is a part of England, or imply that you think Edinburgh is in any way Scotland’s most important city… those common tourist mistakes will cause offence!)

One thing to be aware of is, if you’re used to Londoners, Glaswegians tend to be much more direct and expressive. In general in the UK, the further North, the more directly people say what they mean.

Don’t panic if a Glaswegian says something blunt or confrontational that a Londoner would only ever say to a mortal enemy – it’s usually just friendly banter and a sign they trust you have a good sense of humour too. If the laughing stops and someone starts being very precise and pointed about what they say, that’s when to start being careful.

Update – Mast posted a comment above suggesting practicing with a native to get used to the accent. If that’s hard to do, there are several famous Glaswegian comedians who have medium-strength Glaswegian accents. Look for videos of:

• Kevin Bridges
• Billy Connolly  
• Frankie Boyle (warning – not for the easily offended)
• (fictional character) Rab C Nesbitt (also possibly not for the easily offended)
• Episodes of the sitcom Still Game
• Karen Dunbar
• Limmy

This is a personal experience answer. I am Dutch, so maybe a bit more blunt than you, but my solution works well.

I had that same problem last summer. I had arrived in Edinburgh one day, went to visit Glasgow the next and the first person talking to me was hard to understand.

I explained to the woman that English is not my first language and I had not adjusted to the accent yet. She understood and talked a bit more slowly and that was enough for me.

Mostly when you take the blame on yourself, ‘my understanding of English is not very good’ people understand that you do not want to blame them. And people in the different parts of the UK are used to tourist not understanding them. But they also know that they do not speak much (or anything) besides English, so they do appreciate your speaking their language, however limited.

Funny side note, in the past I have translated for a Scot in France. We were a group of volunteers (with English as only common language) and the guy had to read a list of jobs and ask for the number of people needed. I had been there before and more or less knew the list, at least the regular jobs. And the guy the year before had had a mild Scottish accent, so I was used to the sound.
Many of the other volunteers had never heard anybody with a strong Scottish accent and did not know what to expect.
Specially the Americans in the group did often mention not being able to understand it at all.
Fun and it did improve my English, certainly the understanding part.
I was only 19 or 20 at the time and not yet fluent in English, so I was rather proud it worked.