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Create a scheduled script that runs once per day during the night when there are no other scheduled scripts or tasks.
The supplied time is in UTC so make sure that LM has correct timezone set for the automatic conversion to work.
For testing purposes you can comment out os.sleep. It is needed so the script is not triggered again if the time is shifted.

Code:

os.sleep(30)

require('serial')
port = serial.open('/dev/ttyACM0')
if not port then
  log('port not found')
  return
end
port:flush()

function settime(line)
  local parts = line:split(',')
  if parts[1] ~= '$GPRMC' then
    return
  end

  local hour, min, sec = string.match(parts[2] or '', '^(%d%d)(%d%d)(%d%d)%.')
  local day, month, year = string.match(parts[10] or '', '^(%d%d)(%d%d)(%d%d)$')

  if not hour or not day then
    return
  end

  local cmd = string.format('date -u -s "20%s-%s-%s %s:%s:%s"; hwclock -w',
    year, month, day, hour, min, sec)

  os.execute(cmd)
  return true
end

buf = {}
lines = 0

while true do
  char = port:read(1, 1)

  if char then
    if char == '\r' or char == '\n' then
      if #buf > 0 then
        line = table.concat(buf)
        lines = lines + 1

        if settime(line) then
          log('time updated')
          break
        elseif lines > 20 then
          log('could not update time')
          break
        end

        buf = {}
      end
    else
      buf[ #buf + 1 ] = char
    end
  end
end