Solicitors do not routinely lodge the will anywhere. This is a common problem where solicitors' firms are wound up and original wills lost. I have had to deal with this numerous times (and it has normally resulted in my proving a copy of the Will - which clearly does not exist in these circumstances). The chances of a solicitor realising he has the will and the client may have died are unlikely unless someone does an audit of the strong room.
If a will turns up, the executors of the Will can approach the administrator and they can agree to have the original grant set aside under the Non Contentious Probate Rules and proceed to probate on the later will. If no agreement is possible, there will have to be a High Court Chancery Division action for revocation of the Grant of Letters of Administration.
The time limits that apply really depend on what had happened. If the administrator acted honestly, the time limit is 12 years (s22(1) Limitation Act 1980), but if there is a breach of trust, there is no limitation. The 12 years runs from the "date on which his right to receive the interest arises". A recent case Re Loftus states this is from the Grant, however in these circumstances, there is a reasonable argument for the time running from death.