A 1930s house probably has a rather weak and crumbly oversite, with no DPM. Polythene sheet had not been invented. I bet if you scrape it with a teaspoon you can make a groove.
A new plastic pipe does not have to follow the same route as the old one
I would insulate new plastic and lay it under the floor, drilling or chiselling diagonally down through the concrete and wall to meet the trench you dig in your front garden. Unless it is paved or concreted, this will not be a complicated job in warm dry weather.
When I did one of these I'm pretty sure Thames said it had to be 750mm deep, and they sent someone out to inspect it before I filled in the trench. Mine went over the top of a drainage pipe but he did not grumble about that.
Get as close as you can to the boundary and see what you can reach. I have heard of people scraping out the ground under the pavement to connect it themselves. In my case I had a services pit just inside the boundary, with an old stopcock, and I connected to that. There was a short stub of half-inch pipe but the flow through my new plastic was greatly improved, as
@oldbutnotdead says
had I been less frugal I would have paid TW to make the connection to the meter. If I was getting an unvented cylinder I probably would have done. With the pit just inside the boundary It would not be difficult to do it at a later date, without needing a new trench.
I strongly recommend running it at full size inside the house, all the way to your full-size kitchen stopcock, then teeing off a pipe for the hot (to your boiler, tank or cylinder) and another for your cold taps. This will minimise the risk of uncomfortable showers.