Will I need a work visa for the UK after Brexit?

Talk with HR at your company. Talk with your immediate supervisor, and even ask the CEO or COO, because you won't be the only one asking this question. It's going to depend on them continuing to sign your paycheck. If I understand correctly, there IS no law covering this.  But if there's a "no deal" Brexit, and they say they won't pay you...well. That's what you need to know.

My GUESS is that either the UK will do the "no deal" Brexit, (in which case there will be so many disasters you could go unnoticed in the chaos) or May will unilaterally decide to delay the exit indefinitely, which the EU has said they would allow. Either way you should be OK.

The bottom line is that the UK government has no flippin' idea what it's doing. There IS NO way to satisfy all their legal obligations, and the entire government is operating on a lie ( that Brexit would magically fix everything.)  The UK economy has already fallen behind France, so now the UK is the 7th largest economy, not 6th. It will get worse. They're going to be too busy putting out fires and trying to get food from Europe to worry about work visas.

Expect confusion. 

Here's a piece from the BBC:

"May created a special cabinet department to oversee negotiations with the EU. It instantly disintegrated into “a total and utter shambles.” Officials were left in ignorance of what they were supposed to achieve. “It could be anything … They didn’t really know where to start.” “A big part of the job for officials was educating politicians about the implications of the political narrative they had established.” The cabinet minister May selected to oversee negotiations was described by a civil servant who worked with him: “He thought he knew a lot but most of what he’d written was wrong in some way: legally, diplomatically, or just plain not correct.” "

But for you, it depends on what your employer says and does.