I've been doing exactly this with a Raleigh Arena 10 speed bike.
Its ridiculously expensive to buy new parts for this old bike, so I was fortunate to pick up a crash damaged road bike that had frame damage but the transmission was okay.
You need long-reach brake calipers to move from the 27" wheel to a 700c wheelsize, which is 4mm smaller. I got some NOS Tektro R556 dual pivot calipers cheap, and while they fit the front wheel perfectly the rear still doesn't reach the rims. More work required here.
The donor bike had Shimano 105 3x9 and that has transferred across okay. I was astonished that the cartridge BB fitted right in.
The front derailleur is clamp-on and requires a shim for the smaller seat post. I also had to be creative with the pull cable routing given its bottom pull and the original front mech was a top-normal bottom pull "Spirt"
Rear wheel spacing was also surprising in that the rear wheel fits perfectly, but in the smallest gear the chain binds on the frame. So this needs fiddling.
Up the front end, I chose to stay with my original drop bars. The quill stem stays for now too while I fiddle with fit, but it will likely change to a much shorter one.
You do not need to replace the fork to change the wheels. However aluminium rims will do a lot both for safety on braking and for overall lightness. The old Chromed steel rims didn't brake very well, and got worse in the rain. You can get Quill Stems down to 20mm in length, and it saves the cost of a replacement fork. Getting a safe 1" threadded fork is going to be hard enough, plus the threadded to threadless adapters are a bit of a bodge. I'd get a nice used quill stem off ebay,
I intend on running 35mm tyres, mostly because I have them already and they fit fine.
Not sure about the stubby mudguards. In reality they aren't worth it, but they look so right on the bike. But mine are completely rusted.
Why am I doing this with a bike that isn't valuable or even nice? In the late 1980s I rode a green version of this bike to high school. Stupidly I threw the bike out when aged mid 20s, thinking "I have a car, why do I need a bike?" (yes, really) and I utterly regret this decision now. This blue Arena was in my neighbour's shed and he gave me a bunch of bikes, so I kept it. When the donor bike came up, I knew it was time to progress this restoration.
Note this shows the 105 brake calipers which don't reach. I'm also missing the top nut that your bike has around the quill stem. Don't loose that!
Here's my home-made dropper which was finished today. This question provided the motivation to get on with it.
More photos are found at https://criggie.org.nz/pictures/?gallery=bikes/raleigh_arena/rear-brake-mount