Thank you for your reply @Deva_Rishi!
I rewrote the reading and parsing of the Serial input and this did increase the performance a lot! Using an AP I now get about 100ms for each request-response cycle, which is more than good enough for my use case.
I have kept the server.on part of the code, because I would like the students to be able to dynamically build up the HTML from the Pico code. This way I can pre-program the ESP-01S modules and have the kids specify the name of the AP (and their password), as well as which buttons to display and what message they should send back over Serial.println() to the Pico.
In the setup() function of the Pico, the students can simply send the follwing to set up the buttons and the AP.
+AP;ESP-01S;Password
+B;/ON;ON;ON_OK;50;50;100;50
+B;/OFF;OFF;OFF_OK;50;150;100;50
This will build up the UI that was in the screenshot above. A very simple UI, but a working one 
I am attaching the code (which needs some TLC and refactoring at this point) I will also look over the example you gave above...
#include <ESP8266WiFi.h>
#include <WiFiClient.h>
#include <ESP8266WebServer.h>
#include <ESP8266mDNS.h>
ESP8266WebServer server(80);
String htmlContent = "";
const int led = 2;
boolean isConnected = false;
const byte numChars = 32;
char receivedChars[numChars]; // an array to store the received data
boolean newData = false;
void handleRoot() {
digitalWrite(led, 1);
server.send(200, "text/html", htmlContent);
digitalWrite(led, 0);
}
void handleNotFound() {
digitalWrite(led, 1);
String message = "File Not Found\n\n";
message += "URI: ";
message += server.uri();
message += "\nMethod: ";
message += (server.method() == HTTP_GET) ? "GET" : "POST";
message += "\nArguments: ";
message += server.args();
message += "\n";
for (uint8_t i = 0; i < server.args(); i++) {
message += " " + server.argName(i) + ": " + server.arg(i) + "\n";
}
server.send(404, "text/plain", message);
digitalWrite(led, 0);
}
void setupAP(String ssid, String pass) {
digitalWrite(led, 1);
delay(200);
digitalWrite(led, 0);
delay(200);
digitalWrite(led, 1);
delay(200);
WiFi.softAP(ssid, pass);
Serial.print("IP;");
Serial.println(WiFi.localIP());
server.on("/", handleRoot);
server.onNotFound(handleNotFound);
server.begin();
Serial.println("HTTP server started");
digitalWrite(led, 0);
isConnected = true;
}
void setupWifi(String ssid, String pass) {
digitalWrite(led, 1);
delay(299);
digitalWrite(led, 0);
delay(299);
digitalWrite(led, 1);
delay(299);
Serial.print("Setting WiFi Connection... ");
Serial.println(WiFi.softAP(ssid, pass) ? "Ready" : "Failed!");
server.on("/", handleRoot);
server.onNotFound(handleNotFound);
WiFi.mode(WIFI_STA);
WiFi.begin("Skynett", "frittvilt");
// Wait for connection
while (WiFi.status() != WL_CONNECTED) {
delay(500);
}
Serial.print("IP;");
Serial.println(WiFi.localIP());
server.begin();
digitalWrite(led, 0);
}
void setup() {
pinMode(led, OUTPUT);
digitalWrite(led, 0);
Serial.begin(115200);
Serial.println("Setup complete");
}
void loop() {
server.handleClient();
readFromSerial();
handleNewData();
}
void readFromSerial() {
static byte ndx = 0;
char endMarker = '\n';
char rc;
while (Serial.available() && newData == false) {
digitalWrite(led, 1);
rc = Serial.read();
if (rc != endMarker) {
receivedChars[ndx] = rc;
ndx++;
if (ndx >= numChars) {
ndx = numChars - 1;
}
} else {
receivedChars[ndx] = '\0'; // terminate the string
ndx = 0;
newData = true;
digitalWrite(led, 0);
}
}
delay(25);
}
void handleNewData() {
if (newData == true) {
String buttonValues[8];
int index = 0;
if (strstr(receivedChars, "+B")) {
char* command = strtok(receivedChars, ";");
while (command != NULL) {
buttonValues[index] = command;
index++;
command = strtok(NULL, ";");
}
if (index == 8) {
String returnVal = buttonValues[3];
server.on(buttonValues[1], [returnVal]() { Serial.println(returnVal); server.send(200, "text/plain", returnVal); });
htmlContent += "<button style=\"position:absolute; top: " + buttonValues[5] + "px; left: " + buttonValues[4] + "px; width: " + buttonValues[6] + "px; height: " + buttonValues[7] + "px;\" onclick=\"fetch('" + buttonValues[1] + "');\">" + buttonValues[2] + "</button>";
}
}
if (strstr(receivedChars, "+AP")) {
char* command = strtok(receivedChars, ";");
while (command != NULL) {
buttonValues[index] = command;
index++;
command = strtok(NULL, ";");
}
if (index == 3) {
setupAP(buttonValues[1], buttonValues[2]);
}
}
if (strstr(receivedChars, "+WF")) {
char* command = strtok(receivedChars, ";");
while (command != NULL) {
buttonValues[index] = command;
index++;
command = strtok(NULL, ";");
}
if (index == 3) {
setupWifi(buttonValues[1], buttonValues[2]);
}
}
newData = false;
}
}
