Turnaround time is one of the biggest reasons we attract a lot of transfers. Obviously we are no more immune from resource spikes than any other NB, but SGS really does have a very good record of timely review completion.
(I would give the credit for that to our long-suffering admin staffers who do a fantastic job of scheduling and resource allocation.

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In my experience, when reviews have gone badly off the rails it is rarely 100% the fault of the NB. Even if we completely forget to schedule a review we can often juggle things around to minimise the impact. I don't remember losing a client because of that kind of thing - companies tend to be very tolerant when you can demonstrate that you are bending over backwards to resolve problems.
I think what usually really sends things awry is a lack of clear communication between the client and the NB. If the review requirements / submission requirements aren't clearly understood by both sides, there is a risk that ping-ponging of questions/submissions can take place.
"Deadline-itis" can also throw a huge spanner in the works. Obviously as a NB we need a good idea when the review will start. Whilst we will try to portray the impression that you are our only client, regrettably that just ain't so.
Thus if deadlines are missed, it can then be a longer time before the review can start, since commitments made to other clients need to be kept as well. So there is understandably strong pressure on the RA team to get the submission to the NB on time. Equally the RA will have made commitments to their management about getting the submission to the NB on time.
Unfortunately that can occasionally lead to submissions being sent in before they are fully complete.
Obviously the Regulatory Affairs Dept can say "Well we submitted it on time, it's the NB's fault".
Unfortunately submissions that are in a poor state will usually take MUCH longer to resolve. They are harder to read/understand, and thus take longer to review. That can mean the whole review schedule getting messed about. And of course when the review IS finally completed it will contain a large list of things that need correcting. That in turn puts the whole project back even further. It also causes stress between the NB and company - stress that would have been completely avoidable.
So it is very much in the interest of the company that there is good oversight of the submission process. Getting an independent sign-off of the file before the submission is made can be a good idea. Having the process covered by the Internal Audit programme is also advisable.
(Having worked on both sides of ths fence I like to investigate this area during audits:
"Who approves the dossier completion process as having been fully completed? How is the effectiveness of the dossier completion / submission process monitored?" )
Obviously it is very important for the company to keep in regular contact with the NB. Most NBs will be able to be flexible on start times given reasonable notice, and only send in the submission when it is as complete and understandable as possible.
For very complex reviews, face-to-face meetings ahead of time can be beneficial to both sides.
Your post seems to indicate that your NB simply doesn't have the resource to cope with your demands. If so you possibly made the wrong choice in the first place, or at least didn't keep track of your relationship if things changed over time. Alternatively the resource at the NB changed, either they lost some people, or perhaps they are simply a victim of their own success.
At this point therefore I think you have several options:
1) transfer completely - this is a considerable project of course, but it can be managed, as discussed above
2) transfer partly (e.g. all new products can be certified by a second NB) - there are quite a few companies that have more than one NB - this option is a little bit scary, not least because it can be expensive, but it does give you some flexibility in not having all your eggs in one basket.
3) live with it, but work much more closely with your existing NB to manage future projects in a timely manner. (Obviously this is good advice for everyone...

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In my pre-edited version of this post I put in a footer saying "You may even want to offer more money". I realised I should have expanded that statement to make it clear what I meant.
My point is that if turnaround time is absolutely critical to you, and the costs of review are greatly outweighed by the cost of launch delay, talk to the NB about how you can help fund increased flexibility and/or increased resource.
At the very basic level, most NBs will have set rates, although some may offer premium services for an increased fee, e.g. a guaranteed initial response within a set time. (There are issues with that approach.)
Alternatively you could pay for pre-review sessions. This could be the form of a face-to-face meeting with the reviewer, to discuss the general structure/content of the submission. This would serve to strengthen the final submission.
NBs tend not to like receiving submissions in dribs and drabs, it makes taking an eagle-eye's view of the whole submission very difficult. However in some situations a modular submission approach is acceptable.
Another alternative, potentially expensive, is that you could reserve additional blocks of review time well in advance. The risk is that may never be used by you (due to the submission being late). If the NB can fill those blocks with other work you will probably not be charged, if they can't fill in the time then you would lose your "deposit".
I hope this information helps. Everyone is always welcome to drop me a PM if you need a confidential discussion about your situation/options.