Skipper Lewis Moody and scrum-half Ben Youngs both travelled to New Zealand with knee injuries while Tom Wood and Nick Easter missed the final warm-up win over Ireland with calf problems.
Mark Cueto (back spasm) and Mike Tindall (hamstring) both suffered injury scares in the 20-9 victory at the Aviva Stadium, while Simon Shaw only arrived in Auckland on Thursday after a stomach bug delayed his departure.
However Johnson claims his squad will be at full fitness come Monday.
"Injury-wise they are all progressing well. We expect them all to be fit to train fully at the latest on Monday when we are in Dunedin," said Johnson.
"They are always training, but I mean be there, not restricted in whatever they do on Monday. That is the plan at the moment.
"Simon Shaw got here this morning and he looks very well. They are all OK."
England's arrival in New Zealand led the national news, with around 300 supporters greeting the squad at Auckland airport.
"It was a great welcome," said Johnson. "People talk about World Cups for a long time and to actually get there and see the excitement was good fun.
"It was a nice way to come in. It is like the countdown has started. The whole world of rugby is here."
With the start of the tournament fast approaching, Johnson believes that a mental resilience is needed to win the World Cup.
Debates have been going on in New Zealand over what the All Blacks' Tri Nations defeats to Australia and South Africa will mean for their World Cup chances.
However Johnson insists he will not allow England to get ahead of themselves just because of their own confidence-boosting win against Ireland.
"Everyone wants to speculate on what the results mean," said Johnson.
"I was watching TV yesterday and they were asking people to text in asking 'Has it affected the All Blacks? Is it what they need?'
"Does it affect the All Blacks, does it affect the Wallabies? If they let it (it will).
"I am sure they will be very competitive despite a couple of losses.
"It was the same with our warm-up games. We come in off a good win but when we go into the Argentina game we have to start again and win that game from the first minute.
"When you get to the World Cup it doesn't really matter what happened last game, last month, last year it is about what happens on the day.
"I don't think there is a certain style that will win the World Cup - there is a certain resilience that will win the World Cup.
"You have to deal with the pressure and expectation. The expectation on New Zealand will be huge, the expectation on South Africa as the defending champions, we have expectation from our public.
"World Cups are about pressure, dealing with it off the field, dealing with it on the field when you get into close games.
"Finding a way to win is what World Cups are about. If you talk about style, it is the team that can battle back and find a way to win a close game."
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