Potentially nothing - it's the executor of the estate who has to pay the tax (out of estate funds), not the recipient. He still has to pay the tax even if he'd left nothing.
If we forget quibbling about the details of who pays, £150,000 is the tax on £350,000 - but that £350,000 is additional to the nil rate band of the deceased, and we don't actually know what that was - depends on his circumstances, but the overall value could be over £1m. It could also depend on what other bequests were made - there's no one simple answer.