Depending on how far back Attlee is removed from the timeline, this could make a big difference. If Morrison beats Attlee in 1935, then it's likely that Labour will support a coalition under Halifax instead of Churchill in 1940 (as Morrison, contravening Attlee, wanted to), and Halifax as PM obviously means big differences for WWII.
Skipping forward to 1945, Morrison opposed leaving the coalition and forcing a general election. So if he is leader before VE Day, Labour will stay in the coalition until at least VJ Day, possibly longer. This would allow Churchill to delay things until he is ready (and might concievably weaken Labour as an opposition party), just possibly allowing the Tories to win or at least keep Labour's majority small and limit what it can do in government. There's also a chance that Morrison might be persuaded to commit Labour to a joint platform with the Tories, especially if delaying the election and sticking with the Tories through to the Autumn leads to some bye-election victories for the far left (Morrison seems to have been quite certain that Labour couldn't win in 1945).
If Labour does win and Morrison becomes Prime Minister, his government will be less radical than Attlee's was. I don't think he'll appoint a broad-church Cabinet - no Bevan, no Shinwell, possibly no Ellen Wilkinson, maybe even no Stafford Cripps. The initial raft of nationalisations will go ahead, but the programme will stop dead in 1947. Reforms and modernisations in healthcare, education, social security, etc., will be less sweeping and structured to preserve an even greater degree of private provision and local authority control than was kept in OTL. However, and slightly bizarrely, the house-building programme will be controlled by Whitehall rather than by local authorities, and will slow significantly when industry later complains about the amount of materials going into housing (assuming Morrison's criticisms of Bevan on housing were genuine, rather than just attacks on a rival). There will be considerable disatisfaction mid-way through the Parliament and both Bevan and (with greater chance of success) either Dalton or Cripps, might challenge for the leadership at that point.
Morrison's choice of Foreign Secretary will be crucial, as his own handling of foreign affairs will be disastrous (as it was during his brief tenure as Foreign Secretary). On the one hand, he will be more supine than Attlee was to the Americans; on the other, where there are policies with which he cannot agree, he won't be able to handle arguments as well as Attlee and Bevin. On China in particular, Britain won't achieve the compromise it did - it will either cave in to America, or the "special relationship" will go down. With both Morrison as PM and a more right-wing Cabinet, it's likely we'll try to cling on to India even longer. A civil war will break out, with British troops in the middle of it for an extended period. Whereas the Tories would try to stick things out through a prolongued Indian war, Morrison will probably end up handing the whole thing to the UN to sort out, and India and Pakistan won't be part of the Commonwealth.
On the NHS, Morrison wasn't just a municipal socialist - he was conservative when it came to removing or even undermining existing private providers. It could well, at least in some areas, end up exactly like Medicaid in the US, a means-tested system aimed at ensuring the poorest are able to get healthcare (and generally failing to achieve that aim). Even if local authorities are legally required to ensure health services that are free to everyone at the point of use, this won't be funded by the Treasury, so councils could well be empowered to fund it through insurance rather than the rates (and Tory-run councils will certainly choose to do that). Many hospitals won't be publicly-owned - they will continue to be private providers with the council becoming a commissioning authority (which won't be good). Most importantly, rather than one big battle with the BMA over pay and conditions, it might end up with every local authority having to fight the same fight individually.
All that being said, Morrison was very much the Gordon Brown of his day - a lot of his positions, his attitude to coalition and his disagreements with other members of the Cabinet, were at least partially motivated by his ambition to be leader, in particular the need to undermine any potential rivals. Like Gaitskell, his intransigence over the budget for rearmament in 1951 was really about forcing Bevan to quit and wreck his chances of being leader - and Bevan fell for it. If Morrison actually were leader during the period, his positions might well have been different.