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Professor Lloyd gives an interesting talk, but I have to strongly disagree with the professor's interpretation of his experimental results, because with respect, (imco), he seems to have accepted a false (though widely accepted),  interpretation of Special Relativity, without actually carefully checking the logic of the paper (On the electrodynamics of moving bodies).
Specifically at this point Is Time Travel Possible? Testing the ‘Grandfather Paradox’  
The professor says “... traveling forward in ‘time’ at different rates, [which] happens all the ‘time’, in fact, it’s an intrinsic part of Einstein’s theory of Special Relativity”. But I think this can easily be shown to be incorrect.
If we actually check Einstein’s theory of Special Relativity, i.e the source paper On the electrodynamics of moving bodies” itself,
(the electrodynamics of moving bodies | abriefhistoryoftimelessness)
In Section 1 (kinematics),  *Relativity* itself only actually observes that there is “movement”, and “assumes” there is “time”.
Specifically,  the (translated) paper in actual fact says...
"Electrodynamics" "section 1 Kinematics                                                                                                         
If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time...
If, for instance, I say, “That train arrives here at 7 o'clock,” I mean something like this:
“The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.”
This seems perfectly acceptable, unless you realise that the paper says we compare the co-ordinates ( location) of one thing to a thing called “time”,
But in fact the co-ordinates of one thing (a train) are only compared to the coordinates of another thing ( the location of a rotating pointer).
At this point you really have to think for yourself, logically
-IF a thing called time exists, THEN a rotating point is a useful indicator,
-a “rotating pointer” in no way at all proves there is a past, a future, or a thing called time that flows or that we are ”moving forward in” etc.
So, surely, logically, if Einstein does not actually show that a thing called “time” exists, and must “pass” for things to be able to move, then Professor Lloyd's statement that special Relativity shows we can  *“travel forward in time at different rates”, is, imo, *completely wrong.
Firstly because Special Relativity does not show us that a thing called time exists, (and can be dilated etc). You can see in section one, in Einstein’s own translated words, that the paper only shows us that things exist and can be moving, and that we may choose to call one example of motion “time”...
Secondly because, while Special Relativity does show us that moving things ‘are’ changing at dilated rates, without an accompanying proof that there is a thing called time, this finding can logically, and scientifically, only be interpreted as fast moving things *are changing more slowly than expected*, and (imo) not, in any way at all that fast moving things are “heading into a ‘future’” at different rates.
If the above is correct and understood, then one may see it has rather far reaching consequences for our understanding of what does an does not exist in the universe, and our interpretation of Relativity.
Concerning Professor Lloyd's discussions on ”time travel”, I respectfully suggest (for a number of carefully thought out reasons) they may all be based on the above fundamental misunderstanding, and therefore  moot, and each time travel scenario can be understood without accepting the existence of unobservables ( e.g. “the” past, “the” future, and “time”).
Matthew Marsden
(Auth "A Brief History of Timelessness")
(this youtube explains the essence of my reasoning, based on a time travel lecture by professor Cox.
Time Travel,Timeless Answers to Prof Brian Cox's Science of Dr WHO  
and this one a version of the grand father paradox, or more accurately Hawking’s mad scientist “paradox” Does Time exist? How 'Time travel Paradoxes' can't happen without "the past".