I think May does have the support of her party publically (of course behind the scenes they are fighting - do you really think Boris would have been chosen to be foreigner minister if he and his supporters within the party didn't hold some power) apart from one (Ken Clarke) they all supported government plans on the timeline for triggering article 50. One tory mp (after being criticised for supporting the government) has spoken about the need to have a second referendum Anna Soubry (who was great in the campaign)
Labour's problem is support of the party members which is at the moment overwhelmingly for Corbyn, elected MP's that is not the case and if he really cannot gain control over the infighting over Brexit he will possibly step down rather that have another leadership challenge. There is no doubt Keir Starmer is in the shadows waiting and would be a far better leader one who understands and is extremely experienced in negotiations that is what the labour party need now (it is what the country needs now) not a rebel mp party leader who's expertise is turning up at a rally and rebelling against his own party.
and of course for UKIP it is a win win situation it always will be while Corbyn is leading the party