self.log(.., on_epoch=True) runs extremely slow

Nobody has claimed this yet.

### Bug description

When using
`self.log("train/loss", batch_loss, prog_bar=True, on_epoch=True, on_step=True, logger=True, sync_dist=True)`,
it took ~5s to finish this command, slowing down the training extremely.
When I switched to
`self.log("train/loss", batch_loss, prog_bar=True, on_step=True, logger=True, sync_dist=True)`,
the slow behavior went away.

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

v2.0

### How to reproduce the bug

```python
`self.log("train/loss", batch_loss, prog_bar=True, on_epoch=True, on_step=True, logger=True, sync_dist=True)`
with both wandb and tensorboard loggers enabled.
I found the same slowdown when disabling any of the two loggers.
```

### Error messages and logs

```
# Error messages and logs here please
```
There is no error message but a very slow logging operation.

### Environment

Current environment

```
#- Lightning Component (e.g. Trainer, LightningModule, LightningApp, LightningWork, LightningFlow):
#- PyTorch Lightning Version (e.g., 1.5.0):
#- Lightning App Version (e.g., 0.5.2):
#- PyTorch Version (e.g., 2.0):
#- Python version (e.g., 3.9):
#- OS (e.g., Linux):
#- CUDA/cuDNN version:
#- GPU models and configuration:
#- How you installed Lightning(`conda`, `pip`, source):
#- Running environment of LightningApp (e.g. local, cloud):
```

### More info

_No response_

cc @carmocca @Blaizzy @borda

Reproduce the reported self.log call with on_epoch=True, on_step=True, sync_dist=True, and logger enabled, comparing it with the on_step-only call. Investigate the logging path responsible for the slowdown and define done as eliminating the excessive delay while preserving epoch and step logging, with a regression test for the reported case.

Written by the indexing model from the issue text.