...So if I have in my books a building worth say, £60,000 the book value is £60,000 but the fair value is what the current market price is or what I sold it for or...?
That doesn't sound right as if its in your books you cannot exactly know what you'll get without selling it, unless it is based solely on the market price?
Your last assertion is generally correct; looking at a balance sheet will not typically give you a picture of the fair value of the assets, nor their liquidation values, nor anything of that nature. A balance sheet gives you
book values, which is usually the original acquisition cost of the asset, possibly after reduction by some depreciation or amortization deductions that may have been taken.
Hence, when discussing an asset's book value, you'd probably want to avoid references to "worth" (as in the first quoted statement above), as "worth" generally denotes some version of fair market value, to most people.
Exceptions exist; in some cases--and for some purposes--a balance sheet will be prepared which presents all assets and liabilities at their fair values (or at least a good estimation thereof). But unless you're reading a bal sheet which clearly states that it's prepared on a fair value basis then it's a decent assumption that the numbers are at book value. In such case all bets are off as to whether those numbers are a good indication of their fair market vals. That building on the books for £60K just might have a current selling value in the neighborhood of 60K, or it might have a current value vastly different from 60K.
Remember that accounting financial statements have a number of very important purposes, but they're not intended to be statements of fair value. Hence none of the foregoing is intended as a criticism of accounting practices; on the contrary, I think the accounting process does an amazingly admirable job in achieving those important business objectives. It's just that determining fair values of assets is a different field altogether, and not generally in the scope of standard accounting.
Enough rambling. To the point of your question, the book value of equity as it might appear on a typical balance sheet shouldn't be relied on as an accurate measure of the equity's
market value. Other mechanisms---such as active trading of the shares in a robust market with good information flow, or perhaps professional appraisers---will serve in that "valuation" role.