You won't have to take the carpet up completely. Just lift it of the grip strip along the longes wall an a little round each corner, replace that skirting board, repeat for the next wall and replace the carpet along the longest wall and into the corner where you have replaced the skirting board, do the other wall , put the carpet back leaving the corners where you have the old skirting, do the last wall.
You will need to paint or finish the new skirting before putting it in place, then touch up when it's down
Lastly decide if it looks OK or not. You will probably need to rent or buy a knee kicker and tucker to replace the carpet.
Personally I would not lift the carpet completely as I would not be able to fit it as well as a professional fitter.
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