BOAC
Pilots still have to aim for their CTOT, which will be advised by company or handling agents. ATC at the airfield have the leeway to let the aircraft depart -5 to +10 on their issued CTOT.
They will take this in to account when letting you start and/or clear you for taxying.
Each airfield has a declared taxy time (10 mins seems to be standard except at busier or bigger airfields), and so if you call for start with less than that time parameter till your CTOT, then ATC may refuse you start clearance and ask you to get the company to file a delay (DLA) message. Then it's to the back of the queue by Brussels alas
As a basic rule, pilots need to calculate leaving the stand and building in the average taxy time to the runway in use in order to make the CTOT issued by Brussels. ATC have a little leeway to allow for the miscalculations which inevitably occur.
All I would say is that if you are late on your slot and flying from a UK airfield to an airfield outside the UK and have a CTOT and miss it, then the UK can't do a lot to help you. You may possibly go the back of the queue and incur a big delay. C'est la vie ..... and rightly so.
For internal UK flights subject to UK Regulations, we can work a little magic for anyone not taking the piss