We've been in our house for 30 years now, and having had problems with the Internet over the past week, it has only just come to my attention what a Master Socket is, and the fact that we don't have one of the newer NTE5 ones (that's what Wikipedia says the name of it is!)
Would BT replace our existing socket witha newer one?
If you ask BT/Openreach to visit purely to update the Master socket then I would expect them to charge you for that service.
If they visit to sort out an issue with the external wiring then you could reasonably expect the engineer to update it (it is their equipment after all). BTW there are many versions of the "Master" socket and even 30 yrs ago some form of master socket would have been fitted.
If you were in a position to (and wanted to) upgrade to Infinity then as part of the engineer install the latest Master socket would be fitted.
You will have a 'Master Socket', albeit a LJU1 or LJU1/A. If you had say 3 in your house it can be difficult identifying which is the actual Master as they all they are all the same size. The only way to tell which is which would be to take the front of each and see which one has a capacitor in it as that will be the Master one.
Its not difficult at all, you either;
Pay your CP to pay Openreach to come out and install one.
Pay an independent Telecoms Engineer/Electrician to fit one.
Nip to B&Q or Screw Fix and get the LJU and cable yourself.
If you do it yourself though don't for the love of God use that phone extension cable that can only be described as speaker wire as that certainly won't do your broadband any favours. Use a proper 6 wire or better yet Cat5e cable.