"Reading of a will" was a fantasy of the movies - never existed. If there is a will the executor is the person who takes possession or leave it with he solicitor - executor can decide whether to copy it to others. While the normal practice is to show it to the immediately affected persons in fact only each beneficiary is entitled to see only the piece affecting them.
Assuming there is a will and assuming it is brought to probate then eventually the will can be seen at or a copy obtained from the Probate Office - usually takes at least 12 months from death.