The FA is following Government advice which at the time of writing, does not offer any advice to say that we should not be shaking hands, but highlights and emphasises the need for good hand hygiene at all times.
However, if teams feel happier not to shake hands (or to try an alternative method like fist bumping) then that is absolutely their choice and leagues are encouraged to be understanding of teams making this choice.
Should teams decide not to shake hands or opt for an alternative method, this must be conveyed to the match referee before the teams enter the field of play.
Alternatively, a league may choose to issue a policy to remove the procedural pre/post match handshake, but this would be solely at the discretion of the league.
The FA is actively monitoring and adopting Government advice, and as a matter of good practice would recommend that leagues and clubs adopt the same approach.