As far as they say, yes if the exact same Mobo, BIOS, etc. is swapped out the licensing shouldn't be a issue for Windows. Personally though I think your putting too much at stake on this high risk, as the 'replacement' is probably used, and again how long will it last, a month? a year? then your back in the same boat.
In Risk Management I think this is considered a 'crux' moment, when it would be best to address this high risk and high cost scenario instead of trying to 'push it off' again with another risky solution. If you have expensive software on this computer, you should have the backups and most of all original CD/DVDs for it. Computers and software do not retain their 'value' in how you perceive the 'expense' you originally invested in it. For example a Color Laser printer 10 years ago (probably when you got this laptop and software) was easily $10,000 for anyone willing to buy one. Today you grab one at Walmart for $199 and is Wireless for cripes sake. So while you invested in that $10K 10 years ago, is that still worth 10K today? No. The same with the 'very expensive software'.
It is time to look at this like any other infrastructure (vehicles to move your products, buildings to rent to work out of, manufacturing machine that now is obsolete, etc.) and time to reinvest wisely in the long term use of the software's current iteration (more options? more integration? less install costs? etc.?) as compared to these legacy things. You get one good Nimda or even a Stuxnet (which doesn't need any Internet connection) and this machine is a total loss and you lose ALL the software. Now would be the best time to fully fund properly bringing this system up to date, up to code, and with backup and methods for all case scenarios.