Stump removal is nearly always quoted separately from felling, and the two methods available produce very different results for very different money.
The figures
MyJobQuote puts stump grinding at £60 to £400, driven mainly by stump diameter and by whether the machine can reach the stump.
Access is the hidden variable. Stump grinders range from narrow pedestrian machines that fit through a standard gate to large tracked units. If the only route to your back garden is through the house, the job needs the small machine and takes considerably longer.
Grinding versus excavation
Grinding uses a rotating cutting wheel to chew the stump down to somewhere between 15cm and 30cm below ground level. It is quick, relatively cheap, and leaves a hole full of wood chip that you backfill with soil.
What it does not do is remove the root system. The main roots stay in the ground and decay over years.
Excavation means digging the stump and root plate out entirely, usually with a mini digger. It costs several times more, needs machine access, and leaves a large hole and significant spoil to dispose of. But the ground is genuinely clear afterwards.
When grinding is enough
For most domestic purposes it is. If you want to re-turf the area, plant a border, or simply stop tripping over a stump, grinding is the right answer and paying for excavation is money wasted.
When it is not
If you are building over the spot, grinding is inadequate. Foundations, a garden room base, a driveway or a patio laid over decaying roots will settle unevenly as those roots break down over the following years.
The same applies if you are planting a replacement tree in the same position. Decaying roots harbour honey fungus and other pathogens, and the new tree is competing with a mass of rotting timber.
On South London's clay soils there is a further consideration. Removing a large tree changes the moisture balance in the ground, and the clay can swell as it rehydrates. Where a property has any history of subsidence or heave, get advice before removing a substantial root system rather than after.
Practical points before booking
Locate services first. Grinding is blind work, and drains, water pipes, gas, electricity and telecoms cables running near a stump are all vulnerable. A good contractor will ask you what is buried nearby and may refuse to grind close to an unmarked service.
Ask what happens to the arisings. Grinding produces a surprising volume of chip and soil, often more than fills the hole. Removal may be an extra cost.
Ask how deep they will grind. "Below ground level" can mean 10cm or 30cm, and if you are re-turfing you want enough depth for topsoil.
Doing it another way
Stump killers containing glyphosate will kill a stump but do not remove it, and the stump still takes years to rot down. They are a slow answer to a different question.
Digging a small stump out by hand is realistic for something under about 15cm across. Beyond that the root plate makes it extremely hard work, and the grinding cost starts to look reasonable.