Console output failed to be logged with offline WandbLogger

Nobody has claimed this yet.

## 🐛 Bug

### To Reproduce

https://colab.research.google.com/drive/1k-wWyw-_8P987-Ay7n-rtEZiGU2pIrev#scrollTo=zK7-Gg69kMnG
### Expected behavior

The wandb will save the console outputs in the file `wandb/latest-run/files/output.log`, but no such file when using `WandbLogger`.

### Environment

- Lightning Component (e.g. Trainer, LightningModule, LightningApp, LightningWork, LightningFlow):
- PyTorch Lightning Version (e.g., 1.5.0): 1.7.1
- Lightning App Version (e.g., 0.5.2): N/A
- PyTorch Version (e.g., 1.10): 1.12.0+cu113
- Python version (e.g., 3.9): 3.7.13
- OS (e.g., Linux): Linux
- CUDA/cuDNN version: cu113
- GPU models and configuration: N/A
- How you installed PyTorch (`conda`, `pip`, source): pip
- If compiling from source, the output of `torch.__config__.show()`: N/A
- Running environment of LightningApp (e.g. local, cloud): local
- Any other relevant information: N/A

### Additional context
Wandb related: https://github.com/wandb/wandb/issues/3978

cc @awaelchli @morganmcg1 @borisdayma @scottire @parambharat @manangoel99

Start with the linked Colab reproduction and the WandbLogger behavior in offline mode, then review the related wandb issue for context. Confirm the difference between the expected wandb/latest-run/files/output.log and the current result; done means console output is captured in that file when using WandbLogger.

Written by the indexing model from the issue text.