I've put carpet in vehicles. A lot of it can be cut or formed as you go but templates are like gold, just allow for extra.
If the carpet is jute backed, a spray down of water will loosen it so forming is a snap.
If the carpet is plastic backed or has a pad attached, heat can help but pressure and fastening while the adhesive sets is more typical. You can also fold or roll some plastic-backed carpet to loosen it before installation. All carpet has a weave and weft (backing is laid in two opposing directions) so work it in two directions.
If you have a carpet with dense pad, you may have to make some cuts in the pad to form around tight radiuses (radii ?).
Good luck with the installation!