What do these two characters on cufflinks signify? (Characters identified: 福 and 壽/夀)
A technique for ‘solving’ such a problem without a human translator

I pretend I did not know Chinese, I use 'Chinese hand writing input' on my iPad to mimic the brush strokes in the image you posted with my finger movement. I got a list of suggestions, Use your judgement to see which one is closest to the original image. Among all the suggestions,  '壽' and '福' are the closest.

Advice of which way is ‘up’

familiarize yourself with Radi­cal https://www.yellowbridge.com/chinese/radicals.php you can tell which side is up generally.

The name of the language

Copy and paste whatever words into Google Translate and click 'detect language' button. It will tell you what language the words are in.

Whether Simplified or Traditional

Copy and paste '壽' and '福' into Google Translate and set target result as 'Simplified Chinese', you will get '寿' and '福'. Then you know '壽' is a traditional Chinese characters and '福' is used in both traditional and simplified Chinese.
Finally, check online dictionary to learn the meaning of the character

壽 http://www.cantonese.sheik.co.uk/dictionary/characters/1686/
福 http://www.cantonese.sheik.co.uk/dictionary/characters/1102/