Yes, it is very important that you tell Companies House. You cannot start using a new LLP name until Companies House has approved the change and provided you with a certificate of name change.
You can inform Companies House by post or online through WebFiling using form LL NM01, or you can send notification via software filing through a company formation agent. The latter option is the quickest. Either way, you must pay £8 to Companies House to change the name.
Before you apply, please ensure that:
The name is available to register and is not too similar to another company/LLP name on the register.
Companies House should approve the name change within approximately 24 hours of receiving a request, but it may take longer. When Companies House issues your certificate of name change, you must update your websites and official stationery accordingly.