Update: This new R package [`gam.hp`](https://cran.r-project.org/web/packages/gam.hp/index.html) seems to be the solution I have been looking for: 
It contains basically one function (`gam.hp()`) that takes GAMs and GAMMs constructed with the `mgcv` package and calculates either Adjusted R² or explained Deviance *per each predictor* in the model formula. 
It also calculates which fraction of the explained deviance or R² is uniquely attributable to a certain predictor and the publication reference below discusses in how far that it useful to consider. Computation can take a while for complexer models.
Further details see here:

#### Reference: 

Lai J, Tang J, Li T, Zhang A, Mao L (2024) Evaluating the relative importance of predictors in Generalized Additive Models using the gam.hp R package. Plant Diversity, 46, 542-546. [[Link](https://www.sciencedirect.com/science/article/pii/S2468265924000854)]