A neighbour's weed has become a large tree and keeps sending shoots into my garden

Which is why the actual answer is the awkward one: the tree has to be dealt with, and it is not yours.

Approaching the neighbour works better than people expect if you frame it correctly. Most people do not know what they have. "That grew from a weed and it is a species that damages foundations and drains, here is the identification" is a very different conversation from "your tree annoys me". The risk to their own building and drainage is usually the part that lands.

It is worth being clear that the general legal position in many places allows you to cut growth that crosses onto your property, and that this does not extend to entering their land or killing their tree — and that the specifics vary a great deal by jurisdiction. If it comes to that, the question is one for somebody locally qualified rather than for a forum.

Document everything meanwhile: photographs with dates, where suckers appear, any damage. That record is what makes any later conversation, formal or not, straightforward.

On the mechanics, and this is the hard part: cutting suckers on your side will not work and can make it worse.

A suckering tree responds to the loss of a shoot by producing more, often several for each one removed. You are pruning it, from its point of view, and the root system supplying them is under someone else's garden and is not affected by anything you cut.

What can work on your side of the boundary:

• A root barrier. A physical membrane sunk vertically along the boundary to a sufficient depth. This is real work — a trench — and it is the only purely physical measure that reliably stops root spread.
• Treating the suckers rather than cutting them, which for a suckering species means an approach that reaches the root system. This is where local regulation matters, since what may be applied and where varies, and near a boundary you need to be careful about drift onto a neighbour's land.

Both are treating a symptom while the source remains.

One practical note while this is unresolved: keep the suckers cut regardless, even though it does not solve anything.

A sucker left to grow becomes a stem, then a trunk, and then you have the same tree on your side with roots of its own. The cutting is not treatment, it is containment, and stopping is worse than continuing.

Check along fence lines and under shrubs where they are easy to miss for a season. That is where the ones that become trunks always start.

First step before anything else: identify it. Everything about the answer depends on the species, and the answers differ enormously.

The pattern you describe — extremely fast growth, hollow or soft stems in the first year, bark developing later, aggressive suckering from roots at increasing distance — fits several well-known invasive trees. Some of them are notorious specifically for this behaviour, and some are regulated in various places precisely because of it.

Photograph the leaves, the bark, and any flowers or fruit, and get it identified locally — a garden centre, an extension service, a local naturalist group, or a plant identification community. It takes a day and it determines the whole strategy.

It matters for a second reason: if it turns out to be a species listed as invasive or notifiable where you live, there may be obligations on the owner and support available to you, which changes the conversation with your neighbour completely.