Despite being a biochemist, I'll say that my answer is not based on any empirical evidence and just represents my opinion on the matter:
First, I don't think there distinction between the tail end of the protein and the interior of the protein is very good. Certainly proteins can have thiol (i.e. sulfydryl) groups on the interior or exterior and chelation agents would directly remove mercury from the exterior thiols. However, the tails of proteins are not necessarily solvent accessible, nor would most proteins necessarily have thiols on their tails. So, that's the first issue I have.
Second, is the issue of whether elemental mercury can cross plasma membranes. This is important because proteins on the exterior of cells won't have free thiols. The extracellular environment is oxidizing, so most thiols on extracellular proteins are involved in disulfide bonds. In contrast, cells maintain a reducing environment in their interior, so most of the thiols in cytoplasmic proteins are not oxidized and may be free to bind mercury. Since elemental mercury is nonpolar, I would guess it probably can cross the membrane, but I'm not 100% sure about this.
Third, the ability of chelators to efficiently remove mercury from interior thiols assumes that the affinity for exterior thiols is similar to the affinity for interior thiols. This may not be true. The interior of proteins is nonpolar and nonpolar mercury atoms would likely prefer the interior environment over the aqueous exterior environment. At the same time, the interior of proteins is fairly densely packed, but it is not inconceivable that a small mercury atom (van der waals radius of ~0.15nm compared to the 1-10nm size of most proteins) could fit into the interior of a protein without too much trouble. If the affinity for interior sites is much greater than exterior sites, it will take a much longer time for chelators to remove these mercury atoms. If this is the case, a better analogy would be putting a cloth stained with waterproof ink into water. However, if the affinity of mercury between interior and exterior thiols is similar, the water-ink analogy is perfectly valid. I would assume one could fairly easily do an experiment to see which situation applies to most proteins.