The info at Companies House will be accurate as at the date of the last annual return. You may also be able to find the info in the last filed accounts, but since it is (likely on the info given) to be a private company (rather than a listed plc), then the deadline for filing accounts is the best oart of a year after the year end date.
In terms of his shareholding in the company, it will form part of his estate for the purposes of probate (assuming he didn't die intestate), and only once probate has been completed will the ownership of the shares be definitively amended.
In any event, the information at Companie House is not definitive as to who the shareholders are of a company. Only the register of members in the company's staturoty books will be definitive as to legal (note, not beneficial) title to the shares. The books are a private document, and so not visible to the public (employees fall into this category).
Wills are not public documents, unless they need to be made so for the purposes of probate, so you won't find much informaiton there.
In terms of what happens in the future, you are reasonable in your concerns, but there is little you can actually see that will give you any meaningful informaiton without, frankly, bursting into the reading of the will.
If and to the extent that the shareholding is transferred to holders who do not wish to have the shares, then they will obviously look to transfer them for value. However, do not assume that there is a ready market for the shares, so the process may not be quick, and is unlikely to change the ownership of the business overnight. There may well be pre-emption provisions in the company's articles of association (which you can find at Companies House) requiring shares to be first offered to the remaining shareholders before being transferred to any third party. Even if there are no pre-emption provisions, then the relevant shareholder will still need to find a willing buyer for those shares (which may be a minority holkding, and so having little real value).