Should I transfer funds for the cash-purchase of a 1-bed Flat before the contracts are exchanged?

In the US, at least, it's a wise idea because the lawyer holds the money in an escrow account, and the money transfer would have already cleared the bank. The seller's solicitor would have faith that all the money is with your solicitor, thus eliminating one slowdown.

(Of course, your solicitor might be crooked and needing your cash. Pretty unlikely, though.)

In the UK the two key stages of a property sale are exchange and completion.

At exchange, you will usually transfer a significant sum (often 10% of the purchase price) to your conveyancer. This is your deposit (not to be confused with a mortgage deposit). If you're also selling a property at the same time then your buyer will also provide a deposit, and your conveyancer can use the money from your buyer's deposit to pay all or part of your deposit. At this stage you are committed to the purchase, and should be prepared to lose this deposit should you change your mind.

As part of the exchange you will usually agree the date of completion (which might be the same day, but is most commonly a few weeks later). This is when your conveyancer sends the full funds to your vendor's conveyancer, and ownership of the property legally changes.

Money transfers are not always instant, and in a house sale various parties may want to double check things before a transfer is authorised, which can all add delay. Therefore conveyancers will usually ask for funds a day or two before they are needed.

You've not mentioned the agreed completion date. If there is a significant gap between exchange and completion then the deposit and balance would usually be sent to the conveyancer separately. This avoids unnecessary mortgage interest, or in the case of a cash buyer means the funds can remain in other investments until they are needed. You will not earn interest on money held by the conveyancer.