Yes, you can claim straight away on life insurance. There is a standard exclusion on any claims arising from suicide in the first 12 months applied by most insurers. But other than that any valid claim would be accepted.
For a claim to be processed a death certificate will need to be provided to the insurance company meaning you will have to wait until one has been produced at least.
Short term alternatives
It is also worth noting that some insurers provide temporary cover for a fixed period of time while the application is being processed. This ensures some level of cover while the underwriting process is completed, something that can take several weeks if extensive medical information is required from a GP surgery.