MATL, 45 bytes

737515`tZ'-tkwy-':HH:MM:SS'XOw'%03d'YDwYcDXxT

TIO apparently doesn't support clearing the output, but thankfully MATL Online doesMATL Online does!

At 2 bytes more, a bit more CPU-friendly version that does a pause (sleep) every second:

737515`tZ'-tkwy-':HH:MM:SS'XOw'%03d'YDwYcDT&XxT

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737515 is "31st of March 2019" represented MATLAB's default epoch format - number of days from January 0, 0000, optionally including a fractional part to represent the time of day. (I tried to shorten this by calculating it somehow, but its only factors are 5 and another six digit number (147503), and I couldn't figure out a way to do it in less than 6 bytes.)

`    % start do-while loop
tZ`- % duplicate Brexit date, get current date (& time), subtract
tk   % duplicate the difference, get the floor of it (this gives number of days left)
w    % switch stack to bring unfloored difference to top
y    % duplicate the floored value on top of that
-    % subtract to get fractional part representing time
':HH:MM:SS'XO % get a datestr (date string) formatted this way
w    % switch to bring number of days back on top
'%03d'YD      % format that to take 3 places, with 0 padding if needed
wYc  % switch to bring time string back on top, concatenate date and time
D    % display the concatenated result!
T&Xx % clear screen after a 1 second pause (the pause is optional, without it the call is `Xx`)
T    % push True on stack to continue loop