The 22-year-old was with friends when two men and two women, who were with a child, started behaving aggressively.
The man was then punched, chased and struck with the cleaver on the arm suffering cuts and a possible fracture.
The cleaver was pulled from the child's pushchair, police said. Two men and a woman aged 20, 28 and 22 were later charged in connection with the attack.
Unprovoked attack
The attackers left the scene before police arrived.
"This was a completely unprovoked attack which has resulted in the victim sustaining injuries amounting to grievous bodily harm," said Det Insp Andy Roberts.
"It is shocking to think that not only did the offenders have a meat cleaver while out and about and travelling on public transport, but that it was pulled from a child's pushchair."
Police have asked anyone who was on the tram at the time of the attack, about 2240 BST on Tuesday, to contact them.
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