This is normal with old lead pipes. I strongly recommend that you replace it with new plastic pipe (in a larger size, 25mm or 32mm) all the way from your internal stopcock to the connection under the pavement. Ask your local water co to test your drinking water for lead content BEFORE you start work, there may be a lead replacement subsidy or they may at least reconnect free or replace their own pipe at the same time. Get the request in straight away as there may be a delay.
Any subsidy scheme may be hard to find, presumably due to poor website design or possibly a desire to hang onto their money.
It is easy work for anybody who knows how to use a spade.
Plumbers are weedy little fellows with petal-soft hands so may need a burly woman to dig the trench. It is harder is there is much concrete in the way but a local builder or a moler can do it.
The water co may encourage you to use one of their panel of contractors but this will add to the price and is not obligatory. They will inspect the trench before you refill it to verify it is deep enough.
The new pipe dies not need to follow the same route as the old one.
Replace lead pipe inside the house with copper, at least to the kitchen tap and the boiler.
Use larger diameter new stocks to match the bore of the new pipe (they are more expensive and some plumbers may try to skimp).
You are likely to have greatly improved flow and can install an unvented cylinder for unsurpassed hot water later, if you like. Run 22mm copper all the way to it.