It is okay to have the response as a change score (current - initial weight over time) although some prefer the raw response (current weight over time). The most important thing is that we need to control the initial weight, by using the pretest weight as a predictor, which will make the change-score and raw-response approaches equivalent. Instead of an ANOVA table, please consider a mixed-effect linear regression, on which ANOVA is based, that provides extra benefits of showing coefficients and other statistics. The model specification is likely lme4::lmer(change ~ initial * group * week + (1 + week | ID), data =...). We need to test both linear and nonlinear effects of time, for example by adding I(week^2) + I(week^3), using smoothing curves, or having week as a categorical variable instead of continuous. We should also control for age, sex, height, ethnicity, previous diet, and some other initial conditions that cannot be affect by the diet experiment, their interactions among each other, and their interactions with time and diet group.
We assume that once the experiment starts, the only thing that affects weight change is the diet, although this effect may very group and patient characteristics. However, patients often not totally comply with doctor's instructions. Therefore, the actual diet followed and the group assignment may not perfectly align. To correct this, we need to use the instrument-variable method: Measure the actual diet in place of group above in the main equation and use group as an instrument for diet. See 10 Things to Know About the Local Average Treatment Effect https://egap.org/resource/10-types-of-treatment-effect-you-should-know-about/.
By "significant within group differences" among Diet A and Diet B, I guess you probably mean that time is a significant predictor of weight change among group A and B but not C. By "no significant between group differences", I guess you mean that main effects of group are not significant, which is a good sign of randomized group assignment so that the three groups have the same predicted weight change at week = 0. However, I worry if you have included the starting weight as an extra observation for each patient. This initial weight, which is a condition rather than outcome of the experiment, should be used as a predictor not an observation. This is different from an observational longitudinal study where each repeated measurement should be used as an observation.
The most important coefficients of interests are interaction terms. See my answer on this at How to interpret a nonsignificant interaction effect with significant main effects? and Frank Harrell's interpretation of interaction in regression results. We should plot expected weight change over time by group with confidence intervals. See possible plots at https://freshbiostats.wordpress.com/2013/07/28/mixed-models-in-r-lme4-nlme-both/ and model interpretation through marginaleffects.
