I've been unable to find the full text of the Matrimonial Causes Act 1937 (which would have applied at that time). However this summary seems to confirm your belief that the petitioner would normally need to show that the other party to the marriage had deserted the petitioner:
https://www.perfar.eu/policies/matrimonial-causes-act-1937
I suspect that, since the petitioner would have had to pay the court fees, the court might have been prepared to accept an arrangement whereby the husband said "Yes, you can have a divorce on the grounds that I deserted you and, because of that, I'll submit the petition and pay for it"