I started flying Hawk T1 airtests in the mid-80s and I am sure that the current dive to 0.98M was in the schedule by then. The directional trim change that occurs as you accelerate does vary from airframe to airframe and is associated with the tail cone. If you take the tailcone off, immediately refit it and then fly the dive again it can behave differently so it is very sensitive to rigging. There is inevitably a wing drop during the acceleration that requires about 1/3 lateral stick to maintain wings level although, again, it varies between airframes and there is some correlation with the sideslip due to the directional trim change. The recovery is flown wings level at 2.5g because there is a transonic pitch up during the deceleration and the normal acceleration limit at those Mach numbers is only 4.5g. All of this is fairly classical for a wing of that quarter chord sweep angle. I have to say that on this airtest profile I never felt that there was a particular risk of exceeding 1.0M inadvertently because there was always adequate nose up tailplane authority to reduce the dive angle if the acceleration was too high. However, if an airframe displayed atypical handling characteristics for whatever reason I could see how it could happen.
I have intentionally been to about 1.05M but I am intrigued to know how HS ever achieved the 1.2M limit!