In Gaussian regression, the MLE for the coefficient vector is equivalent to the OLS estimator, and the MLE for the error variance has its usual form relating to the residual sum-of-squares in the regression. That is, it can be shown that the MLE is related to the goodness-of-fit statistics by:
$$SS_\text{Res} = \sum_{i=1}^n (y_i-f(\mathbf{x}_i, \hat{\boldsymbol{\beta}}_\text{MLE}))^2
\quad \quad \quad \quad \quad
\frac{SS_\text{Res}}{n} = \hat{\sigma}_\text{MLE}^2.$$
Consequently, we can write the maximised log-likelihood for the model as:
$$\begin{align}
\hat{\ell}_{\mathbf{y}, \mathbf{x}}
&\equiv \max_{\boldsymbol{\beta}, \sigma} \ell_{\mathbf{y}, \mathbf{x}} (\boldsymbol{\beta}, \sigma^2) \\[12pt]
&= \ell_{\mathbf{y}, \mathbf{x}} (\hat{\boldsymbol{\beta}}_\text{MLE}, \hat{\sigma}_\text{MLE}^2) \\[12pt]
&= - \frac{n}{2} \Big[ \ln (2 \pi) + \ln (\hat{\sigma}_\text{MLE}^2) \Big] - \frac{1}{2 \hat{\sigma}_\text{MLE}^2} \sum_{i=1}^n (y_i-f(\mathbf{x}_i, \hat{\boldsymbol{\beta}}_\text{MLE}))^2 \\[8pt]
&= - \frac{n}{2} \Big[ \ln (2 \pi) + \ln (\hat{\sigma}_\text{MLE}^2) \Big] - \frac{SS_\text{Res}}{2 \hat{\sigma}_\text{MLE}^2} \\[8pt]
&= - \frac{n}{2} \Big[ \ln (2 \pi) + \ln (\hat{\sigma}_\text{MLE}^2) \Big] - \frac{n}{2} \\[8pt]
&= - \frac{n}{2} \Big[ \ln (2 \pi) + \ln (SS_\text{Res}) - \ln(n) + 1 \Big]. \\[8pt]
\end{align}$$
As can be seen from this equation, the maximised log-likelihood is fully determined by the residual sum-of-squares and the number of data points. It can also be written in terms of other goodness-of-fit quantities if preferred. In particular, taking $s_Y^2 = SS_\text{Tot}/df_\text{Tot}$ to be the sample variance of the response variable, we can write the residual sum-of-squares in terms of the coefficient-of-determination as $SS_\text{Res} = (1-R^2) df_\text{Tot} s_Y^2$. Consequently, we also have the alternative form:
$$\hat{\ell}_{\mathbf{y}, \mathbf{x}}
= - \frac{n}{2} \Bigg[ 1+\ln (2 \pi) + \ln \bigg( \frac{df_\text{Tot}}{n} \bigg) + \ln (1-R^2) + \ln (s_Y^2) \Bigg].$$
This latter form shows that the maximised log-likelihood is fully determined by the coefficient-of-determination, the sample variance of the response variable, and the number of data points. Once you have an expression for the maximised log-likelihood, it becomes trivial to get corresponding expressions for the AIC and BIC, to wit:
$$\begin{align}
\text{AIC}
&= n \Bigg[ 1 + \ln (2 \pi) + \frac{2k}{n} + \ln \bigg( \frac{df_\text{Tot}}{n} \bigg) + \ln (1-R^2) + \ln (s_Y^2) \Bigg], \\[12pt]
\text{BIC}
&= n \Bigg[ 1+\ln (2 \pi) + \frac{k\ln(n)}{n} + \ln \bigg( \frac{df_\text{Tot}}{n} \bigg) + \ln (1-R^2) + \ln (s_Y^2) \Bigg]. \\[6pt]
\end{align}$$
Asymptotic analysis: If we were to hold the number of model terms $k$ constant and take $n \rightarrow \infty$ we have the asymptotic equivalence:
$$\text{AIC} \ \sim \text{BIC}
\sim n \Bigg[ 1 + \ln (2 \pi) + \ln (1-R^2) + \ln (s_Y^2) \Bigg].$$
Under broad convergence conditions we also have $1-R^2 \rightarrow \sigma^2/\sigma_Y^2$ and $s_Y^2 \rightarrow \sigma_Y^2$, which gives the asymptotic equivalence:
$$\text{AIC} \ \sim \text{BIC}
\sim n \Bigg[ 1 + \ln (2 \pi) + \ln (\sigma^2) \Bigg].$$
