Borthwick, 36, left his role at the Championship side after only 43 days.
The former England captain arrived at Bristol several weeks into the season having helped Eddie Jones coach Japan.
"It's kicked us quite hard because this has been about planning and waiting for Steve to arrive," said Robinson.
Borthwick was named forwards coach with Jones' new-look England in acrimonious circumstances last week, with Bristol owner Steve Lansdown describing the Rugby Football Union as "unprofessional" in their handling of the appointment.
Robinson was forwards coach during England's World Cup-winning era under Sir Clive Woodward before taking over as England head coach himself from 2004-06.
And, while he is unimpressed by the manner of Borthwick's exit, he is certain England have made an astute appointment.
"I was delighted to have signed Steve and I certainly hoped he would be here for longer than 43 days," Robinson told BBC Radio Bristol.
"He is a quality coach and he will be a quality coach for England and I wish him all the best. I know the England team will enjoy working with him.
"I'm disappointed in the circumstance of how it's happened and why it's happened but legally I can't say too much and we have to move forward."
Robinson, who has also coached Scotland, says he will not rush into appointing a replacement and has urged his forwards to handle the responsibility of Borthwick's departure.
"It's down to the players to step up collectively over the next four months for us to achieve what we want to achieve," he said.
"We have to focus on the next couple of weeks, with tough games coming up against London Welsh and Nottingham. There is a lot to be done over the next few weeks."

The company has been negotiating with the pension regulator and the PPF in a bid to offload its pension liabilities and a decision is expected soon.
Most, if not all, members of the scheme would see a reduction in benefits.
Those under retirement age would receive an immediate 10% cut in their pension pot.
In addition, annual payments to most members, including those already retired, would be less in the PPF than if the Hoover scheme continued.
At about £500m, the pension scheme is roughly the same size as BHS.
BHS pensions is already receiving compensation from PPF which acts as a lifeboat for pensions so people are paid even when a company goes bust or is about to and if there are insufficient assets in the scheme.
BHS is being assessed for take-over by the organisation.
The Hoover scheme has a deficit of about £250m.
Parent company Hoover Candy stopped producing washing machines in Merthyr Tydfil in 2009, blaming competition in the marketplace.
More than 300 workers lost their jobs although there is still a head office and a distribution warehouse in the town.
It is believed about half of the Hoover pension scheme's 7,800 members live in Wales.
Pensions expert Stuart Price, partner at Quantum Advisory, said: "Members of the pension schemes will lose out on their benefits but, on the flipside, those still employed by the company will retain their jobs.
"At the moment they [members] don't know what's happening so they can't really plan for their short-term future.
"If they get a 10% reduction [for those under 65] and lower increases in the future they might find it very hard to overcome."
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Bristol director of rugby Andy Robinson says the club have been hit hard by the departure of forwards coach Steve Borthwick, who he believes will be successful with England.
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Up to 4,000 people in Wales could be affected if the Hoover pension scheme goes into the Pension Protection Fund (PPF), BBC Wales understands.