Can I nest LightningModules inside child modules?

Nobody has claimed this yet.

### Bug description

Suppose I have a `LightningModule` (parent) that contains a `nn.Module` (child), which in turn contains another `LightningModule` (grandchild). Calling `.log` inside the `LightningModule` (the grandchild) results in the following warning:
> You are trying to `self.log()` but the `self.trainer` reference is not registered on the model yet. This is most likely because the model hasn't been passed to the `Trainer`

The trainer is only set on the direct `children` of the parent `LightningModule`, not all the descendants, since the `trainer.setter` uses `self.children()` rather than `self.modules()`: https://github.com/Lightning-AI/pytorch-lightning/blob/3730e980e388c23f7e9d1f535793e8d614633362/src/lightning/pytorch/core/module.py#L221-L226

### What version are you seeing the problem on?

master

### How to reproduce the bug

```python
# %%

import lightning as L
import torch
from torch import nn

class GrandChild(L.LightningModule):
def dummy_log(self):
self.log("foo", 1)

class Child(nn.Module):
def __init__(self):
super().__init__()
self.module = nn.Linear(1, 1)
self.grandchild = GrandChild()

def forward(self):
self.grandchild.dummy_log()
return 1

class Parent(L.LightningModule):
def __init__(self):
super().__init__()
self.child = Child()

def training_step(self, batch, batch_idx):
return self.child()

def configure_optimizers(self):
optimizer = torch.optim.Adam(self.parameters(), lr=1e-3)
return optimizer

def train_dataloader(self):
return torch.utils.data.DataLoader(
torch.utils.data.TensorDataset(torch.randn(10, 1)), batch_size=1
)

# model
parent = Parent()

# train model
trainer = L.Trainer()
trainer.fit(model=parent)
optimizer = parent.configure_optimizers()

loss = parent.training_step(batch=None, batch_idx=0)
```

### Error messages and logs

```
You are trying to `self.log()` but the `self.trainer` reference is not registered on the model yet. This is most likely because the model hasn't been passed to the `Trainer`
```

### Environment

Current environment

* CUDA:
- GPU:
- NVIDIA A100-SXM4-80GB
- available: True
- version: 12.1
* Lightning:
- lightning: 2.2.1
- lightning-utilities: 0.11.2
- pytorch-lightning: 2.2.1
- torch: 2.3.1
- torchmetrics: 1.3.2
- torchvision: 0.18.1
* System:
- OS: Linux
- architecture:
- 64bit
- ELF
- processor: x86_64
- python: 3.11.9
- release: 5.15.0-113-generic
- version: #123-Ubuntu SMP Mon Jun 10 08:16:17 UTC 2024

### More info

_No response_

cc @carmocca @justusschock @awaelchli @borda

Start at src/lightning/pytorch/core/module.py around the trainer.setter referenced in the issue, then run the supplied Parent, Child, and GrandChild reproduction. Check how trainer registration behaves for nested modules and add regression coverage showing that the grandchild can call self.log() without the warning.

Written by the indexing model from the issue text.