The origin of Bolthole
Rob Pierre has been working on turning the Dolists around ever since he seized power in May, 1905 PD. War of Honor begins in June, 1918 PD, just over 13 years later, and the battles involved take place in late 1920 PD, more than 15 years later. The technology transfers leading to Shannon Foraker's innovations were already in the pipeline well before Eighth Fleet's offensive brought on the ceasefire, and Pierre began construction of the Bolthole Yards in 1909 PD, four years after the need for something like them became apparent and eleven years before the first ships built there actually saw combat.
The initial yard was intended primarily to provide reserve building capacity in an area safe from Manty attack because the Manties didn't know it existed. (Think the Russian tank factories moving across the Ural Mountains.) It was built under the same sort of security the US practiced for the Manhattan Project, because the whole point was for it to be secret. It was also built to employ the most modern construction techniques then available to the PRH, and once it was in place, the fact that it was a secret installation made it a logical place for Pierre to transfer his various secret R&D programs to. The initial work on designing the PRH's missile pods, for example, was done at Bolthole, and it was also the place where most of the initial reverse-engineering of Sollie tech was undertaken all along.
So by the time Theisman shoots Saint-Just, this massive yard complex is already up and running.
8th Fleet attacks Barnett on Christmas Day, 1914 PD. Saint-Just is shot on May 10, 1915 PD, 3 years before the beginning of War of Honor. From the outset, the Havenites realize what the Manties did to them and how the SD(P)s have to be designed. The fact that someone else has already put them into production, and the fact that the RHN (only it was the PN then, but so what?) has actually seen them in action means that they have a pretty fair notion of their general design and capabilities, rather than having to come up with the initial concept on their own. They already have the pods. Building the pod launching capability into their ships isn't all that difficult once you recognize the need to do so, and they did not attempt to match the other sophisticated features of the RMN designs. They basically just hollowed out an existing SD design and wrapped it around the pod rails. Moreover, the SD design they used as the basis was already a much more capable class than anything the Manties had yet seen, incorporating a lot of reverse-engineered Solly tech and other improvements which provided much of the platform for Shannon's later accomplishments. (Do not make the mistake of assuming that Shannon is the only competent tac witch/tech witch in Havenite service. She may be the best of the lot, but there are others.) So, basically, they spend a year at crash priority designing the launch rails into the ships, then spend 2.5 to 3 years building the first units of the class, and they're able to deploy them in time for War of Honor. Note, too, that all throughout the book it's pretty apparent that they're still ramping up production when they actually pull the trigger. What they used in their initial operations was basically the first surge of a pretty impressive tide of new construction.
As far as the LACs go, there is no major new tech in their designs at any point except for the improved missile launchers. Remember, too, that in RMN service, the new missile pods were developed from new launcher technology for LACs; the Havenites simply reversed the process and developed the improved LAC launchers from the existing pod technology when they recognized the need for it. Aside from the new EW capabilities built into them (again, a product of the long-standing Havenite programs to reverse-engineer Solly tech into their fleet as a whole), the entire point of the Havenite LACs is that they are very austere designs whose capabilities come from a ruthless suppression of everything not necessary in an already existing class. And the PRH (unlike the RMN) was still building LACs as of the start of the war and so began with a more "up to date" platform as the basis for Commander Clapp's development program. As for the CLACs themselves, as Janacek (among others pointed out) designing them isn't particularly difficult if you have the LACs to put into them in the first place. And unlike the RMN, which was basically testing a new concept (and fighting entrenched opposition every inch of the way), Thomas Theisman's navy was never in doubt of the need to acquire LAC and CLAC capabilities of their own. Not to mention the fact that they'd had a victim's-eye view of just how devastating they could be in combat. Like the German armored theorists post WW2, they had an enormous motivation to analyze what the Manties had done and find ways to compensate for or even improve upon the Manticoran techniques again, using existing technology.
Economically, again, Pierre had been working on addressing the economic problems of the PRH all along, although it was only in the last 5 or 6 years of his rule that he really made major progress. The biggest problems, however, were always concentrated in the core planets of the PRH -- those which had been part of the PRH's disastrous welfare system for the longest and had the largest Dolist populations. The initial funding for Bolthole was basically scraped up by canceling BLS increases and actually reducing them gradually (there are numerous references in the books to the fact that even the Dolists were willing to accept this after Cordelia got them convinced that it was necessary to defeat their "class enemies") and diverting that funding to naval construction and the development of the yard at Bolthole. In addition, there were other systems (and Bolthole was one of them) where the "Dolist mentality" had never become so ingrained in the first place and the local economies were much stronger. Remember, some systems had to be showing a positive balance on the ledger sheets, or the drag of such economic sinkholes as Haven would have destroyed the PRH years ago, conquests or no.
("Bolthole," by the way, is simply the code name assigned to a system which already had a fair population -- well up in the hundreds of millions -- before the yard was moved in. The system in question, however, was "off the charts" as far as foreign intelligence agencies were concerned for several reasons. The main one is its extreme distance from Haven -- there's a reason Theisman can only get out there once or twice a year. It's way far on the other side of the RH's space from the SKM, and in an area which, overall, was never noted for its industrial or economic muscle before, in essence, the CoPS moved in and established its "Five-Year Plan" to turn the system into an industrial powerhouse. And, BTW, they never even attempted to give it any other industrial capacity. Basically, this is an entire star system which is one, huge production line for warships and an R&D base and nothing else at all.)
The degree to which the Havenite economy has flowered following the cease-fire is largely the result of the removal of highly inefficient central-planning bureaucracies and the ruthless diversion of available funds from the maintenance of the welfare system into a program deliberately designed to generate reinvestment and development even at the expense of considerable real suffering among the largely defanged Dolists of the Mob (those that were left after "Citizen Admiral Clusterbomb" got done with them, at any rate). Indeed, one reason for the "Pierre reforms" (which Pritchart has continued) was to force the Dolists to become productive or starve.
The fact that so very many Havenite citizens were poised to grab the economic ball and run with it says a lot about the "gray economy" which was already running and about the "dead hand" effect of the old, Legislaturalist state. It also, unhappily, confirms Pierre's long-standing belief that if the Legislaturalists had been willing to bite the bullet, admit the failure of their system, and introduce internal reforms much earlier, they probably could have solved the PRH's problems without ever turning Conquistador. The real problem, all along, was that they didn't think they could do that without surrendering their monopoly on control of the state and facing the consequences when they were called to account for their catastrophic, self-serving mishandling of that state.
At any rate, Pierre & Co. got the yard up and running even before he put through his last package of financial reforms. What happened under Pritchart and Theisman was that they simply maintained an existing program, put one of their best and most trusted people in charge of it with orders to adapt its output to reflect the new tactical realities demonstrated by 8th Fleet, and funneled funding generated by the reviving economy (but never formally accounted in their public budgets) to pay for it.
IMO, this represents a task which required enormous effort, but not one which was inherently impossible in any way. Remember, however economically crippled the PRH may have seemed to the people who actually had to run it and fight the war with what it could build/provide, if it had been as hopelessly crippled in fact as you seem to be assuming above, it would have collapsed under the pressure of the Manticoran Alliance long before 8th Fleet was required.