Before I start, I realize that this question is related to another question. However, this question is more concerned about the custom of wearing one's "Sunday best", which usually means a nicely brimmed Sunday hat and modest-looking dress. This has been portrayed in literature and in movies. Example: Chicken Sunday, written by Patricia Polacco, and Bridge to Terabithia (2007 movie). Is this solely an American Christian custom? Are there examples in other Christian cultures? How did this custom come about, and how did it evolve over time?
I am unsure as to most other Christians. Among those that I have studied with and continue to study with they have always dressed formally.
I believe that Christians of any denomination dress nicely when going to worship god. It's part of giving god your best. Here in an article that may explain a bit better and I think it is relevant not only to witness', but all Christendom. Why dress up for meetings
This doesn't mean your clothes had to be expensive. As long as you are clean and presentable then that is acceptable.
It seems as far as history goes Christians not only wore, but where encouraged to wear simple clothing so as to not distract from the meaning of the occasion. This is especially true in the 18 and 19th centuries. Worshipers usually dressed plainly with no ornate jewelry and such. This is not only for the reason stated above, but that your everyday person didn't have the means to come by these things.
Based on a little more research it seems dressing up was a social status thing and then later became a part of representing and honoring god.
In the UK, customs vary wildly between churches. Catholic, High Anglican, Methodist and far-right baptist traditions tend to promote, at least implicitly, smart dress but with no particular dress codes. Pentecostal traditions tend to promote the relationship aspect of corporate worship, seeing church as a place for people to come into God's presence in safety and support, and so avoid the idea of "Sunday best" - often supported by 1 Samuel 16:7 (NIV):
7 But the Lord said to Samuel, ‘Do not consider his appearance or his height, for I have rejected him. The Lord does not look at the things people look at. People look at the outward appearance, but the Lord looks at the heart.’
There is always a caveat that people should be dressed modestly - in a way that is not sexual or overly distracting from the main point of worship: communion with God. Some churches do follow a dress code very similar to that seen in America - for example in Brethren traditions. But these days, very few churches would pass comment if a person arrived shabbily dressed, and and in many churches jeans and tee-shirt might be considered the norm!