There are two separate permission regimes and a tree can be caught by either, or both. People check for one and assume they are clear.
Regime one: a Tree Preservation Order
If a TPO applies, you need written consent from the local planning authority before cutting down, topping, lopping, uprooting or wilfully damaging the tree. Government guidance also treats cutting roots as requiring consent.
Apply to the council. There is no fee for a TPO application. Expect around eight weeks for a decision, and be prepared to justify the work with an arboricultural report if you are asking to fell rather than prune.
Regime two: a conservation area
This one catches far more South London gardens than people expect, and it is the one most often missed.
If a tree stands in a conservation area but is not covered by a TPO, you must give the council six weeks' written notice before doing the work. This is a section 211 notice. The council can grant consent to proceed earlier, but if it does nothing you must wait the full six weeks.
The six weeks exist so the council can decide whether to make a TPO. Sometimes it does exactly that, which is a legitimate outcome rather than an obstruction.
Very small trees below a diameter threshold set in the regulations are exempt from the conservation area notice. The threshold is measured at a specified height above ground and the detail matters, so ask the council's tree officer rather than estimating it yourself.
How to check
Call the tree officer at the relevant borough: Southwark, Lambeth, Lewisham, Wandsworth, Bromley, Croydon, Greenwich, Merton or Sutton. Most publish TPO and conservation area maps online, but a phone call gets you a definitive answer on both regimes at once.
Ask two explicit questions: is this tree subject to a TPO, and is this address in a conservation area. A negative answer to the first tells you nothing about the second.
When you do not need permission
A tree in your own garden with no TPO, outside any conservation area, and not subject to a planning condition or restrictive covenant, is generally yours to remove.
Two caveats worth knowing. New housing developments frequently carry planning conditions requiring specific trees to be retained, which bind you even without a TPO. And some South London estates, the Dulwich Estate being the obvious example, impose their own covenants on top of anything the council requires.
The exemptions that do apply
Dead trees can be removed without consent, but you must give the authority five working days' written notice first.
Work is exempt where the tree presents an immediate risk of serious harm and the work is urgently necessary. This is narrower than it sounds and you have to be able to prove it afterwards. Photograph the tree before work starts, and get your tree surgeon to put their assessment in writing on the day.
Who carries the risk
The tree owner does. If a tree surgeon fells a protected tree on your instruction, you are the one the council prosecutes.
That is why a competent tree surgeon will ask you about TPOs and conservation area status before quoting, and will decline the work until it is confirmed. Treat that question as a mark of a professional rather than an inconvenience.