Help Required To Automatically Reconnect to BLE Browser

Sorry, it looks like that's not implemented in UART.js, however you could look at using the Puck.js library which is very similar.

In it there is a connection.reconnect method which should do what you need:
https://github.com/espruino/EspruinoWebT­ools/blob/master/puck.js#L236

So you may not even have to change the file, you can do connection = Puck.connect() to get the connection, then connection.device.addEventListener... and connection.reconnect()