Before play started Friday both Broad (knee) and Finn (shin) were ruled out of the opening two fixtures in this three-match series, placing a doubt over their fitness for next month's Champions Trophy one-day tournament in the UK.
"They are key parts of our full-strength one-day team and they are world-class bowlers," Cook told reporters.
"The guys coming in haven't got that experience. They are quite hard boots to fill," he added after Chris Woakes's six overs cost 45 runs and Jade Dernbach's 10 went for 55.
"Clearly it's not ideal but we've got a good medical team and fingers crossed."
Friday saw England add former Ireland seamer Boyd Rankin, now with county side Warwickshire, to their squad for this series, which continues with the second ODI at Southampton on Sunday.
Cook said the 6ft 7in (2.01 metres) Rankin's height had worked in his favour.
"When you've got five bowlers you want as much variety as you can. When you lose two tall bowlers, you want to try to replace them with another tall bowler," Cook explained.
What really cost England dear at Lord's was a below par total of 227 for nine.
"I think we said after 20 overs 260 or 270 would be a par score on that wicket," Cook said.
"We pretty much got ourselves into a position to do that, at 120 (in fact 117) for two."
But three wickets then went down for just nine runs and Cook added: "Obviously losing those three wickets quickly took the sting out of our innings.
"It's like a 'nothing' score, isn't it, 230?"
New Zealand, after James Anderson's first over double strike left them reeling at one for two, eventually won with 19 balls to spare on the back of Martin Guptill's unbeaten hundred.
"When you are defending a small total, you know the only way to win is to bowl them out," Cook said.