How to estimate the variance of regressors in scikit-learn?
I believe it is the probabilistic nature of a model that allows you to get the variance of predictions, or more generally defined as the uncertainty of predictions, like the Gaussian process you mentioned. This is not simply avaialble in standard regressors.
I think you should be looking at Probabilistic regressors like BayesianRidge if you would like to estimate the uncertainty of your model. An implementation  is also avaialble in scikit-learn, also this nice python package based on PyMC3 or directly via PyMC3 itself for instance. In the latter there are examples like for Bayesian regression in Jupyter Notebook with a good explanation.
In principle, Bayesian Models do not return a a single estimate for the model parameters, but a distribution that make it possible to make inferences about new observations as well as to examine our uncertainty in the model. You may find this post useful.
Note: Adding a normal prior on the weights as it is done in Bayesian regression, one turn the Least-Squares problem to regularized L2 regression under the hood as well (see the full math. derivation here).
Updated Answer: I totally forgot the classical yet simple and powerful Bootstrap Sampling method to calculate confidence intervals for machine learning algorithms. A textbook definition says:

Bootstrapping is a nonparametric approach to statistical inference
that substitutes computation for more traditional distributional
assumptions and asymptotic results. A number of advantages:

The bootstrap is quite general, although there are some cases in which it fails.
Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate
inferences when the data are not well behaved or when the sample size
is small.
It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.
It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).

Reference: Fox, John. Applied regression analysis and generalized linear
models. Sage Publications, 2015.

Please note you do not need a model with probabilistic nature. See this post, or this answer or this one.