There's a difference between civil weddings and religious weddings. A religion doesn't typically recognise civil marriages - in the eyes of the Church, a couple is only married if they're married by the Church's religious rites, because marriage is a sacrament.
But a religious wedding isn't legal unless the wedding certificate is signed, which is the same certificate signed in civil weddings.
Margaret could have married Townsend in a legally recognised civil wedding. She could even have married him through another Christian church. But because the monarch is Head of the Church of England, Royal marriages are C of E, the C of E wouldn't have recognised the marriage. Strictly speaking, this means that formally the Queen couldn't have formally recognised the marriage either.
Hence Churchill's concerns. The pressure came from Churchill, who was an old man at that time.
In practice both the Queen and the country would have found a way to accept a civil, legal, marriage between Margaret and Townsend...but this would have meant Margaret relinquishing her Royal status for official purposes.
We have seen in recent years, just how important royal status is to members of the Royal family so I have no doubt that her reluctance to give up her Royal position was instrumental in Margaret's decision to not marry Townsend. I also don't doubt that she'd have continued to be treated Royally in private.