it's easier if you have one person in the kitchen listening, and another person at the external stopcock.
Yes, it is easiest to run a new pipe, all the way to the external stopcock (and preferably to the connection, meter or cock under the pavement or road). It does not have to follow the same route as the old one, it may be easier to trench under a path, or enter under the wooden floor or something rather than dig up the kitchen floor. I recommend running it in 32mm blue plastic, which is pretty sure to be bigger than your old pipe, this will give improved flow for better filling of baths and less chance that one tap will weaken when another is turned on. You need full-bore stopcocks which are bigger and more expensive than the 1/2" ones, so make sure the plumber fits them and does not scrimp. It is a hundred times more expensive to dig up and change later than to do it while the trench is open. You can take a 22mm copper pipe tee off for the cold supply, and another to go to your hot cylinder, water tank or combi boiler, and one in plastic for your outhouses or garden taps. You will be amazed and delighted at the improvement in flow. You can turn it down a bit if too strong.
Plumbers are weedy little fellows with petal soft hands, so a sturdy woman or a builder is more suitable for digging the trench. You can have it moled if necessary. A plumber can probably connect it to the incoming supply, but if it is in the road you will normally have to pay the water company or one of their approved contractors. They might do it free or at a reduced rate if you are replacing lead piping or having a water meter fitted, or if they know you have a long-term leak. If you have any lead piping, ask them to test your drinking water for lead content BEFORE you start any work.
"Blue water supply MDPE should be laid in a trench at a depth of at least 750mm (measured from the pipe crown), but no more than 1350mm, from the finished ground level and at least 350mm away from other utilities (such as gas or electricity)."
and the water co will (should) inspect to verify this before you refill the trench. You must tell them in advance.