How to reconstruct a scikit-learn predictor for Gradient Boosting Regressor?
There are two estimators i.e. The initial predictor and the sub-estimators

init_estimator
The estimator that provides the initial predictions. Set via the init argument or loss.init_estimator.
estimators_
ndarray of DecisionTreeRegressor of shape (n_estimators, 1)
The collection of fitted sub-estimators.

Prediction after the first (i.e. init) estimator is controlled by the learning rate.
You can get the prediction as done in the below code -
trees = model.estimators_

x  = x_test.iloc[10,:].values # A sample X to be predicted
y_pred = model.init_.predict(x.reshape(1, -1)) # prediction from init estimator

for tree in trees:
    pred = tree[0].predict(x.reshape(1, -1)) # prediction from sub-estimator

    y_pred = y_pred + model.learning_rate*pred  # Summing with LR
y_pred