Under the NEC3 contract Option A - the Client has delayed a key date (as a result of a third party) and resulted in the Contractor remaining on the project for a longer period of time. Can the delay in achieving the milestone be a CE and is the Contractor able to claim for the cost of remaining on the project as well as the time?
Yes.
If you will not meet a Key Date due to a reason that is a valid compensation event then as part of the quote not only will you be assessing the date to which the Key Date should be moved to (and possibly Completion Date as well if they are linked), but also the associated costs you will have incurred as a result of that Key Date being later. As a Contractor make sure you have notified (and within 8 weeks otherwise you may be time-barred) that the third party issue has caused you a delay and this is a reason within clause 60.1 for it being a compensation event.
The idea is that with any compensation event you are not is a worse position than you would have been had the compensation event not occurred.