Putting under pressure is mostly about repetition. If your setup is different on the eighteenth green than it was on the first, your stroke has to compensate, and compensations fail when your hands are shaking.
The five checkpoints
1. Ball position
Slightly forward of centre for most golfers, so the putter is moving level or fractionally up at impact. Too far back and you deloft the putter and drive the ball into the ground.
2. Eye line
Over the ball or fractionally inside. Check it with the ball drop test or a mirror.
3. Shoulders square
The single most common error. Open shoulders start the putter left. Lay a club across your chest and look in a mirror.
4. Hands under the shoulders
Not reaching, not crowded. This keeps the putter swinging on a consistent arc rather than being manipulated.
5. Weight slightly forward and still
A stable base. Any lower body movement in a putting stroke is noise.
Build a routine, not a checklist
Five checkpoints is too many to think about over a putt. Work through them in practice until they are automatic, then reduce your on course routine to one thought. Most good putters have exactly one, usually about pace or start line.
How to make it stick
Five minutes before every round with a reference on the ground. Not a technical session, just a calibration. You are proving to yourself that your setup is where you think it is before it matters.