Could also steal a page from the original Master of Orion. You select an emphasis (though MoO did allow you to focus everything) but your research is benefiting all fields to some extent. And once tech progresses, you need to improve the infrastructure of your planets. Of course in MoO it was just a slider and the game even adjusted it automatically for you.
That is roughly how Rule the Waves technology works too... you can select to focus more or less on different technologies. It is pointless to focus on everything as that is the same as leaving everything on normal focus.
You then also commit part of your economy to it. The higher this value is the less you get out of it.
I think that having a "slider" or just selecting a "%" box of the number of wealth you dedicate for technological upgrades for each colony would suffice. Then each colony could have a progress meter for each research category you want to catch up on that goes up over time depending on how much wealth you spend on improving it.
You could also give a lump sum of wealth that then speed up the process by up to twice the speed...
For example...
A colony is generating 1000 wealth and have 100 factories... you get the research to add +20% extra industry power. You have selected 10% of the wealth to go toward upgrading technology at that colony which meas it automatically add 100 wealth per year towards upgrade. Let's say each factory need 10 wealth to upgrade so you would need 1000 wealth to upgrade them all to the new standard. Let's say you produce 1000 points of construction at the start. After one year you have built 10 new industries and spent 100 wealth on upgrades... obviously all new industries is of the new standard... so now 20 of 110 industries are done... but the game does not have to track exactly how many just the overall progress, so the progress now is 18.2% so the colony generate 1139 construction points with 110 factories.
The game just show a progress made at 18.2% to max tech capacity for industry.
If you say... give the colony 300 wealth for upgrades these get stored at the colony earmarked for technological upgrades and for each cycle the colony upgrade it matches money from this storage so it upgrades twice as fast... so... in the above example it would apply 200 wealth the first year and take 100 wealth from this pool to do the upgrades and instead of ending up at 18.6% you instead end up at 30/110 at 27.2% or 1159 construction points
This should be a relatively simple yet effective system that is easy to overlook and will not require much micromanagement.
There could always be a default value for how much money would go towards technological upgrades and you don't have to fiddle with it unless you want it to change. Obviously no wealth is used if there is nothing to upgrade.
Such a system will have to make decisions on how fast you tech up depending on how fast you are able to integrate said technology to your society. You also should not be able to choose which sector to upgrade just a sum to upgrade and it apply that across all sectors that it can upgrade, that is the simplest way to do it. But you can boost it by adding funds and colonies that generate more wealth also can upgrade faster which to some extent make sense as an economically stronger colony should be able to upgrade faster.
Wealth upgrades probably should work slightly different though, at least for the civilian economy.
Or some such system that is simple yet effective and give you a choice of how to utilise your economy and to what benefit. If you already is overtaxing your wealth economy with military assets you might not afford technological upgrades... but then you have not been kind to your economy in the first place.