This is always due to some issues with understanding what is required by the http basic authorization, or how that is done in a particular environment.
As always I recommend using the curl utility if possible and sending some simple requests in if possible. Curl will handle the authorization set for you (you just pass in your API key and a colon e.g. curl -u MY_API_KEY: ... ) and you can switch on verbose mode and see exactly what is being sent and returned.
Companies House have previously suggested some posts in this thread:
Others have apparently managed to get Zapier working with Companies House:
As far as I’m aware there are no issues of Companies House API ignoring the Authorization header.
As far as being blocked - you should read the Companies House documentation on rate limiting and perhaps search this forum for more details e.g. the following thread:
I don’t use Zapier so can’t advise on the details of this myself.