I've run a test on serial performance on a Uno and an ESP8266 with the code below. No UDP involved, so not a direct comparison with your program. The code sends 125,000 bytes of data and prints the elapsed time.
On both the Uno and the ESP8266, there was no latency that I could see. Both of them output 125,000 bytes in just over 5s at 250,000 bits/second. So, well above the output rate that you need.
However, on the ESP8266, the program hung on about one attempt in four or five. So, the serial (or my USB-serial converter) is not bullet proof.
But on your problem, where the amount of serial output is much less than this test, I wonder if the interaction with handling network traffic might be causing problems? I saw this on the Github page for the ESP8266 core:
Remember that there is a lot of code that needs to run on the chip besides the sketch when WiFi is connected. WiFi and TCP/IP libraries get a chance to handle any pending events each time the loop() function completes, OR when delay(...) is called. If you have a loop somewhere in your sketch that takes a lot of time (>50ms) without calling delay(), you might consider adding a call to delay function to keep the WiFi stack running smoothly. There is also a yield() function which is equivalent to delay(0).
Might be worth testing with a call to yield() added into the for loop in your program.
Test program:
#define BLOCK_SIZE 500UL
#define NUM_BLOCKS 250UL
char buffer[BLOCK_SIZE] = {0};
void setup()
{
Serial.begin(250000);
for (uint16_t i = 0; i < BLOCK_SIZE; i++)
{
buffer[i] = (char)('A' + (i % 16));
}
pinMode(13, OUTPUT);
digitalWrite(13, HIGH);
uint32_t startTime = millis();
for (uint8_t i = 0; i < NUM_BLOCKS; i++)
{
for (uint16_t j = 0; j < BLOCK_SIZE; j++)
{
Serial.write(buffer[j]);
}
Serial.println();
}
uint32_t endTime = millis();
digitalWrite(13, LOW);
Serial.print(endTime - startTime);
Serial.print("ms for ");
Serial.print(BLOCK_SIZE * NUM_BLOCKS);
Serial.println(" bytes");
}
void loop()
{
}