If the normal model isn't a reasonable choice for your response variable, first think if there is any suited model. If there is one, the use it. If not, you may only bootstrap the distribution of the statistic of interest (e.g. the mean differences). But with n=6 this is actually not really worth it. This is the problem with bootstrapping: it works well just in those cases where the sample size ia large enough to make deviations from the assumptions largely irrelevant.

Just jumping to a rank-based analysis may be expedient. The meaningful interpretation of results obtained from rank-based analyses requires good knowledge about the distributions - knowledge you won't have and that you won't get, particularily not from small samples. Just for instance you can have two populations with mean(A) < mean(B) and at the same time median(A) > median(B). It's a matter of sample size to get both, the t-test and the the rank-sum test, significant. What now would you conclude? A>B or A

1) The first thing I would make sure you are assessing the normality assumption correctly. It is not the case that the data should be normally distributed. But the conditional normality of the data can be assessed by looking at plot of the residuals from the analysis (q-q plot or histogram).

2) I recommend against using a hypothesis test like Shapiro-Wilks to assess normality in this situation. Looking at plots will serve you better for this application.

3) If the main problem is the heterogeneity of variance, there are various ways to address this for a one-way anova. One is to use Welch's anova. There are other options, and what you use may depend upon what options are available in the software you use.

4) Anova is somewhat robust to deviations from normality.

5) For a one-way design, there are various options for parametric, nonparametric, and robust options. Be ware that different tests may test different hypotheses. For example, Kruskal-Wallis does not test for differences in means.

6) My advice to get good advice here: I would conduct a one-way analysis of variance, and share a) the histogram or q-q plot of the residuals, and b) a plot of the residuals vs. the predicted values. Or, instead of b), a plot of the observed values vs. the groups.

Agree with Jochen Wilhelm

Sample size is too small to conduct a quantitative study.

You should use non parametric test

Since ANOVA is a general regression, whole sample size may be more important than per group sample size. Of course the most important number is error degrees of freedom. In your case,

Sample size: 36

Groups: 6

Total DF: 35

Regression DF: 5

Error DF: 30

From this point of view, your sample size is not very small. Many animal and cell studies have per group sample size 5-7.

Non-parametric tests usually gives you larger P values in such sample sizes.

If fail to predict the outcome linearly, I usually prefer to generalize my model instead of using non-parametric tests.