I have not used TeamCity Server, but one possible alternate solution is to combine the three build scripts into one script. And put tasks of the three separate scripts into separate targets in the master build file. So, instead of the three separate build scripts, you have one build script with three targets, namely deployUI, deployServices, deployBackend. Untested sample below:
<?xml version="1.0" encoding="utf-8" ?>
<Project xmlns="http://schemas.microsoft.com/developer/msbuild/2003" DefaultTargets="DefaultTarget" ToolsVersion="3.5">
<Target Name="DefaultTarget">
<CallTarget Targets="deployUI" ContinueOnError="false"></CallTarget>
<CallTarget Targets="deployServices" ContinueOnError="false"></CallTarget>
<CallTarget Targets="deployBackend" ContinueOnError="false"></CallTarget>
</Target>
<Target Name="deployUI">
<!-- Put UI deployment tasks here -->
</Target>
<Target Name="deployServices">
<!-- Put Services deployment tasks here -->
</Target>
<Target Name="deployBackend">
<!-- Put Backend deployment tasks here -->
</Target>
</Project>