You can (and most likely should) stick to datetime xAxis type, just add proper timestamp data to your series data like so:
const weatherData = data.weather;
const temperatureData = weatherData.map(entry => ({
x: new Date(entry.timestamp).getTime(),
y: entry.temperature
}));
const pressureData = weatherData.map(entry => ({
x: new Date(entry.timestamp).getTime(),
y: entry.pressure_msl
}));
You can further adjust the xAxis labels display based on the API: https://api.highcharts.com/highcharts/xAxis.dateTimeLabelFormats
Here is your demo working well: https://jsfiddle.net/BlackLabel/c62uy7gz/
Kind regards,