Temp sensor with timer and LED

I need to make a sketch that uses temp. [450°] to start a timer for 25 minutes. When the timer starts, it will illuminate an Green LED, signaling that the timer is running. After 25 minutes has elapsed a red LED will then illuminate stating that time is complete.

I wish to use an LCD to display real time temperature as well as lapse time.

I will be using Arduino Mega 2560 board.
Any help and other devices needed will be greatly appreciated.

Have you so far got your temperature sensor working and displaying its value in the serial monitor? Nobody can help you with that until you specify which sensor it is.

Have you bought an LCD screen already?- if not, I'd strongly suggest you get an I2C one which uses only 2 pins to run it (apart from power of course), much simpler than the run-of-the-mill ones which use may more wires.

Firstly, this is a commercial oven reaching to 500°. So I am currently using a "J" Thermocouple wired into a Solo temperature controller.
I am trying to reuse this setup because I have to place the probe onto the part in the oven to get actual temperature.
The LCD is a 16 x 2 display with 2 power connections.

Most of your requirements are trivial to code. The tricky bit will be figuring out how to talk to the temperature controller to get the temperature.