not really. a sturdy woman who knows how to drive a spade can do most of it. Plumbers are weedy little fellows with petal-soft hands so a small local builder or labourer usually does it.
The new pipe does not have to follow the same path under the house.
In your case, as you are end of terrace, it could sensibly be trenched to the side of the house, and enter through the kitchen floor.
The old pipe was quite likely 1/2" ID nominal, which is fairly small but adequate for topping up a loft tank. modern practice is to run a plastic pipe of 25mm or 32mm OD which will give much better flow, sufficient for a decent shower from a combi or a powerful one from an unvented cylinder, and will fill a bath in reasonable time. Ample flow should prevent the squeals of annoyance from someone in a shower when a tap is used elsewhere in the house.
i think your supply pipe is probably iron, but if there is any possibility that the house has lead pipes, ask the water co to test the drinking water for lead content, which is harmful. They may offer a lead replacement subsidy or at least free reconnection to the main. Also ask if they have a leaking pipe replacement scheme. They might connect free if you get a water meter and the new pipe is already in position ready to connect. They will want to inspect it before the trench is filled in to ensure it is at the correct depth. If you have a plumber, ensure they fit full-bore stopcocks at the boundary and inside the house. They are more expensive, but will not impede flow. It is a hundred times more effort and expense to replace stopcocks afterwards, than at the time of laying.
If, as I suspect, the clay drains are cracked and leaking, this is again work for a small local builder, though it is quite a pleasant outdoor DIY job in summer. Look out for wild tomato plants or red worms in the soil which are signs of broken sewers and soilpipes. When reinstating you can lower the ground against the house, or fill the trench with cobbles and pebbles which do not allow water to rise by capillarity. All this can be done in one job.
You mention damp patches on the carpet.
This is consistent with a water leak in the solid floor.
Is there a water meter?
An experienced wrinkly old plumber can confirm a leaking pipe in other ways. It is common.