I think it's unlikely that there was any sort of regular service from New Street to St Pancras by the late 1950s. At that time we travelled regularly between my parents' house in Birmingham and my grandparents' house near Harringay station in north London. Had there been such a service, it would have been much simpler to use that and then a suburban train from KX to Harringay with just a short walk from there, rather than going to Euston, Piccadilly Line tube to Finsbury Park, bus to Stroud Green then a longish walk from there. My father was something of an enthusiast who enjoyed plotting unusual routes by train, and had such a service existed at a suitable time, I'm sure we'd have used it at least once just for the experience, if not regularly. Of course that doesn't rule out a service operating at an impractical time, but I never heard any mention of such a possibility. Once the Euston electrification work began, we initially switched to travelling via Banbury, then my parents bought a car instead.