Share transfers are not reported immediately and are included within the next CS01. Please see How Do I Transfer Shares? for help processing a share transfer.
There is no need to inform Companies House of a share transfer at the time it takes place. Instead, details of any share transfers should be included in the next company's confirmation statement, and it is not until this has been processed that the public record will be updated to reflect the new shareholder position.
However, if you wish the public record to reflect a share transfer without delay, this is easily achieved by filing the confirmation statement. It is quite common for companies to file a confirmation statement early following a share transfer and doing so can give both parties (the seller and purchaser) comfort that the transaction has been "officially" recognised in the public records. This remains a decision for the company however, and there is no requirement to do this. Please see How do I include a shareholder list on the Confirmation Statement?
Using Inform Direct, you can complete share transfers with minimal time and effort. You can also produce share certificates, branded with your logo and pre populated with all the transfer details for the new shareholders. The company registers will be automatically updated to reflect the change in shareholders and when the time comes to file your confirmation statement you will find all the details of the share transfer are automatically included, so you won't need to enter these again.