The 2nd at Conwy is a shortish par three, and it does cause hold ups, particularly in 4-ball play and with parties of visitors. It is a very good hole (by David Williams) but if you don't hit and hold the putting surface recovery shots are particularly difficult. The 6th, the next short hole, also causes hold ups. In fact, now that I think about it, all four short holes can cause hold ups.
At Wilmslow, however, the first short hole is the 6th and on the back nine (the 10th is a legitimate alternative starting place) the first short hole is the 14th, so play flows well. However, as the 14th is a brute of a hole for higher handicappers, it is not unusual to find a log jam on this tee.