Constructive mathematicians definitely do believe the reals exist and are uncountable. Not all reals are computable or definable. The set of computable reals

Constructive mathematicians definitely do believe the reals exist and are uncountable. Not all reals are computable or definable. The set of computable reals is countable. There are different varieties of constructive mathematics. A very large majority of constructive mathematicians believe the reals are uncountable. In intuitionistic mathematics the reals are uncountable. In some constructive versions of math you get weird things like there being an injection from R to N but there not being a bijection due to diagonalization. I think you are confusing computable/nameable with constructive as that term is used by most mathematicians. Note that Chaitan's Constant is not computable.