Short: you'll need to need to request this (see and reply to the thread below) OR you'll need to use the API Company Officers endpoint or the XML Gateway.
Will it be free to pull this data ...
Using the API is free, the XML gateway is chargeable, I don't know about the Officers dataset mentioned above. For bulk data tasks Companies House recommend you use the bulk data products rather than the API. I don't know what data fields are in the bulk data set - presumably company number is listed with each officer so you can cross-reference this with the list of companies registered in the last 2 years from the company bulk product. If not you'll need to use the API and this will take some time at the standard rate allowance of 2 per second (you can look up all officers of a company by company reg. number).
You've specifically asked about the director's names - if you want to find out "who controls" the companies then the PSC list might be more useful. You can get this via the API PSC endpoints OR via a bulk download at http://download.companieshouse.gov.uk/en_pscdata.html
The various data sets / interfaces available are listed at: https://www.gov.uk/guidance/companies-house-data-products
(The officers dataset seems to be only available by asking in the forum.)