Advice needed for Weather station up grade to Internet connection.

Hi.

I have built a Weather Station using an Arduino UNO R3, c/w a BME 280 Temperature, Humidity & Pressure sensor, a BH1750 Lux meter sensor and a UV30A Ultraviolet sensor.

I would now like to get the Arduino to talk to the Weather Station software (Weather Display) so that i can record trends etc; Does anyone have experiencing doing this, i would appreciate all the help i can get.

I have no experience of this Weather Display - the Weather Station software but it should be possible.
A large number of commercial weather stations are supported so you simply have to find out how the data has to be formatted (API description). You should be able to send data to the site through your WLAN. Look at running an ESP8266 to do that and maybe migrate your existing application to such a device.