The full Laws of chess, including those applying to quickplay finishes can be found on the FIDE site .

My comments on

can be found elsewhere.

I have gleaned the changes below from the BCF and/or FIDE sites. The comments are however my own and should not be regarded as in any way definitive, particularly as I do NOT have access to any guidance on these changes given to arbiters.

Recording moves
Law 8.1 now includes:
It is forbidden to write the moves in advance, unless the player is claiming a draw according to Article 9.2 or 9.3.
9.2 and 9.3 are rules governing claiming draws by threefold repetition and by 50 moves without capture or pawn move respectively. In these no move is made on the board, but it is necessary to record the move (if any) that would create the appropriate position.
This change has been highlighted on the BCF site, so they clearly intend to implement it rigourously. It removes the opportunity for the unscrupulous player to use their scoresheet as an aide memoire.

Don't take the King!
Law 1.2 now includes:
Leaving one‘s own king under attack, exposing one‘s own king to attack and also ‘capturing‘ the opponent‘s king are not allowed.
Maybe specifying that Kings may not be captured is purely a tidying-up exercise. However in the absence of knowledge it is fun to speculate. There are some players who like to indicate that their opponents have moved into check, or failed to move out of check by 'capturing' their king. With such action now strictly forbidden they would presumably be liable for illegal move penalties - 2 minutes to opponent and third illegal move means loss of game - as well as having to satisfy the touch-move requirements.

Mobile Phones etc
Law 12.2b:
It is strictly forbidden to bring mobile phones or other electronic means of communication, not authorised by the arbiter, into the playing venue. If a player‘s mobile phone rings in the playing venue during play, that player shall lose the game. The score of the opponent shall be determined by the arbiter.
Clearly it is not practical to ban the bringing of mobile phones into the venues of most competitions, but the rest of the rule should be noted. In particular some may be disappointed to discover that a loss for their opponent may not translate into a win for them.

Law 4.4d:
If a player promotes a pawn, the choice of the piece is finalised when the piece has touched the square of promotion.
So you can safely handle other pieces without being required to promote to them.

Recording with less than 5 minutes
Law 8.4:
If a player has less than five minutes left on his clock at some stage in a period and does not have additional time of 30 seconds or more added with each move, then he is not obliged to meet the requirements of Article 8.1. Immediately after one flag has fallen the player must update his scoresheet completely before moving a piece on the chessboard.
The above is from the FIDE site and appears unchanged. The BCF reports a change to 8.4a:
Once a player has been relieved of the need to score, he remains relieved of this duty until after a flag has fallen.
At first sight both versions are slightly strange. However reading the rest of the rules you realise that the 'proper' approach to time controls is to use the allotted time and then check that you have completed the right number of moves. Digital timers mirror this by not adding time for subsequent time controls until a previous time period has elapsed. With analogue clocks the tendency is to adjust the clock as soon as the requisite number of moves needed for the first time control are completed rather than waiting for a flag fall.
I suspect the intention of this law and of its possible replacement is to avoid the possible argument that might arise if a scoresheet had to be updated immediately the correct number of moves had been completed, since people in time trouble do not always know exactly how many moves they have made.
I am convinced that the rules are not designed to allow players to avoid recording for the rest of the game even though the practice of clocks being put back before a flag fall means that the next flag fall will denote the end of the game. Thus I would suggest that when using analogue clocks the correct interpretation is that adding extra time to the clocks ends a players relief from the necessity to keep score, and that they must then update their scoresheets before making any further moves.