Most of the project management tools or software are for technical projects. Most technical projects are complex and the software is designed to handle the complexity. In contrast, the projects of HR consultancy are non-technical and you may not require complex tools. Against this backdrop, I feel that project management templates in MS Excel should suffice your purpose. I found the following link on Google:
If your projects involve database management, then you may opt for MS Access.
If you wish to have online access to your data, then you may create the templates on Google Sheets. A few links are as below:
The advantage of using Google sheets is the easiness of linking the sheets. If you link the sheets, then the master sheet can be linked to the specific project sheets. As you update the project sheets, the master sheet gets updated automatically. Secondly, if you create hyperlinks, then the data can be accessed for the past 'n' number of years!
From India, Bangalore
However, most of the standard tools can be used for the purpose. You must be able to tweak your processes to match with the software.
MS Project is probably the most popular tool. However, it was designed for a single person, so its collaborative capabilities are a little limited. However in most cases it will work.
Recently, Trello has become popular for such things. Check out their site and their videos to figure out if it meets your requirement
Sometimes, just a plain task management tool like Microsoft ToDo may be enough
From India, Mumbai