One option, if you believe your certificate is lost and never to be found again, is to
request a duplicate. It's an incredibly important document to have for your records if you cannot find the original. It is, however, quite costly to apply.
Another, cheaper, option is to
file a subject access request to the Home Office and either request general SAR as per the options available to you, or ask just for the information you require (e.g. the exact date you became a British citizen) by explicitly stating that this is what you require.
Both options will take time to process. As far as I am aware, there is no expedited way that you can obtain the information you seek through reliable and formal channels.
A faster, but less reliable and not necessarily a option I would personally choose, way is to contact the council where you had your citizenship ceremony. They may hold their own records, though I am not sure if this is standard practice or if they will be willing to do a search.
