Approval of the word British in your company name will depend on how it is used. Normally the Secretary of State would expect the company to be British owned. You would need to show that the company is pre-eminent in its field by providing supporting evidence from an independent source such as a Government department or a trade association.
If the word 'British' is qualified by words that do not describe an activity or product, for example by using a 'made-up' word, then evidence of pre-eminence is not necessarily essential. However you would be expected to show that your company is substantial in relation to its activity or product and that it is eminent in its own field.