It seems like you're saying that covariate B is correlated with predictor A. In that case, that is not a situation where you can use an ANCOVA. In an ANCOVA your factor B would have to be correlated only with the RT, not the other predictors. If you ever do find a situation where an ANCOVA is appropriate it would just be...
data.aov <- aov(RT~ B + A + Error(subject/A), data = data)
If you use the following notation: $sgr_{ft}$ is the growth rate for fish $f$ in tank $t$, $temp_t$ temperature in tank $t$ and $mass_f$ the mass of fish $f$ then you estimate a regression of the form:
$sgr_{ft} = \beta_{0t} + \beta_1 temp_t + \beta_2 mass_f$ using the R-code lmeModel<-lme(sgr ~ mass + temp , random=~1|tank). Note that the intercept can be different for each tank ($\beta_{0t}$ has $t$ as subscript).
The $\beta_{0t}$ can be extracted from lmeModel using the extraction functions fixef and ranef. In fact, $\beta_{0t}$ has two components, i.e. $\beta_{0t}=\beta_0+b_{0t}$, where $\beta_0$ is a fixed effect and $b_{0t}$ a random effect. The function fixef gived you the maximum likelihood estimator of the fixed effects, the function ranef gived the best linear unbiased predictors (BLUP) of the random effects, you will see that ranef yields one value for each tank.
You may also try a model like $sgr_{ft} = \beta_{0t} + \beta_{1t} temp_t + \beta_2 mass_f$, where the coefficient $\beta_{1t}$ also depends on the tank, using the code lmeModel<-lme(sgr ~ mass + temp , random=~1+temp|tank)
Using the R-code lmeModel<-lme(sgr ~ mass + temp , random=~1|tank/temp) is similar to the first model, only that in the the first case you assume a different intercept for each tank while in the model with random=~1|tank/temp you assume a different intercept for each temperature within each tank. So it is something like $sgr_{ft} = \beta_{0t,T} + \beta_1 temp_t + \beta_2 mass_f$, where T in $\beta_{0t,T}$ stands for temperature. However, if you treat temperature as a factor (i.e. a categorical variable) then I think it does not make much difference.
EDIT 27-07-2016:
About the question in your comment; if temperature is categorical, having four values, then this categorical variable is replaces by three (the number of categories minus 1) dummy variables, $D_1, D_2, D_3$.
In that case lmeModel<-lme(sgr ~ mass + temp , random=~1|tank) estimates $sgr_{ft} = \beta_{0t} + \beta_{11} D_{1t} + \beta_{12} D_{2t} + \beta_{13} D_{3t}+ \beta_2 mass_f$ while lmeModel<-lme(sgr ~ mass + temp , random=~1+temp|tank) estimates $sgr_{ft} = \beta_{0t} + \beta_{11t} D_{1t} + \beta_{12t} D_{2t} + \beta_{13t} D_{3t}+ \beta_2 mass_f$
Best Answer
There is a list of tutorials on this subject here:
http://www.r-statistics.com/2010/04/repeated-measures-anova-with-r-tutorials/
Good luck.