I've seen the internet slang
TL;DR many times in the internet, and as I can see people used it pretty much in the present day.
Can I use it in a formal email to a client?
You can use the CONCEPT of a tl;dr in a formal mail. just don't NAME it that. Call it "summary" or a similar term. Clients will love a short and to the point conclusion, because it means they don't need to read a 50 line email if they can't or don't want to. If they want to know more, they can read the rest, but if they are preparing for a meeting or are running low on battery power on their iPhone, they aren't interested in a small novel.
Just remember to be polite in the summary. Don't say anything in there that can be seen as offensive to another party, even if they don't read all of it. But that goes for all formal correspondence.
Anecdote to explain that last part: A blog on weird code and situations developers encounter has a story about a developer who in their mail said something like (including linebreaks added by Jannet's client):
Something is wrong in the development platform. The code written by Jannet needs to be beaten into submission. I'll do that tomorrow.
Jannet saw a line starting with her name and understandably thought she was being threatened with violence. Fortunately, they could get the problem resolved the next day.
I imagine that my internet and email usage is significantly higher than for many of my generation but despite being familiar with most shorthand, I've not previously seen TL;DR . My point is simply that, in a formal letter, you probably want to be certain (not merely "reasonably sure") that the person to whom you are writing will know what you mean. And you don't want them to be in the position of having to do a Google search to discover your message ... and now I'm about to do the search myself!