What are specific strategies to help someone who does not do well under pressure
Not sure about the specific department about which you speak and in reality most departments and supervisors vary according to organisational policy and personal/relational differences. Having said that however, my understanding of the Ph.D.process is very different from undergrad studies for both the student and supervisor. 

Optimally [and again this depends on egos etc.] your supervisor will encourage a collegial/peer based relationship with you whilst providing support and guidance which will diminish over time as you become more independent, assured and gain ownership over your work. In this scenario being timed should not be a factor.

However, this implies a sense of responsibility in your actions leading to academic/professional independence - and at least the potential for this has to be apparent to your supervisor and peers from the beginning of your tenure. In this respect, I would suggest the following:-


Approach your supervisor and discuss the issue in a  solution oriented manner. When doing this try to avoid an attitude of 'well this is just the way I am so suck it up';
Seek out intervention programs at your organization that might assist; things like time management courses, project planning, communication skills and if applicable anxiety management/boosting your confidence therapy/courses.


By being proactive about this issue you are setting boundaries with your peers which can provide breathing space for you thus allowing you to relax into the process and hopefully fulfill your responsibilities with respect to deadlines/time frames and expected output.

**However!**By doing this it is incumbent upon you to take responsibility for the above, find ways to manage what ever it is that causes you to feel as you do and perform as expected. Having a good contingency plan for when things stuff up would also be a good idea. As is being able to re-calibrate if you fall off the wagon so to speak. As they say the best way to deal with a mistake is to learn the lesson and move on. Try not to get embedded in the problem.

And lastly... hopefully not but you may need to acquire some negotiation and/or conflict resolution skills .... sometimes we come up against people who simply don't/wont/cant understand your position and try to impose their regulations/values/modality onto you... 


Avoid such people if you can; 
Negotiate 'normal interactions' with them carefully and in a non-reactive manner; 
Seek advice immediately if they impose.


Hope this helps :-)