I'd go for the gravel then but would look out for free hard-core first.
Bit concerned that your bed is very narrow so you might want to ask about what to plant before you get soil. If you can only grow smaller planrs they need less soil depth so you could vary the depth, maybe deeper in the corners for a shrub or small tree.
I think if you start a new thread, maybe planting suggestions for narrow raised beds, it will get more replies. I can see grasses in there, festuca is a lovely evergreen one but quite short. There is some agapanthus, which doesnt mind being restricted. At the back in your last photo the mauve plant might be a salvia. With the white flowers no idea what they are but maybe ground cover roses, there seems to be lavender in there too.
They are 34cm wide, bit more wriggle room in the L corners.
Calculators I have used so far come out at 2 bulk bags of gravel and 6 bulk bags of topsoil to fill both (one on each side of the garden) 😳
But my question is does 15cm gravel and 40cm soil sound correct. Should I be just using topsoil/mixing with compost?
I think you're not far off. Ben Cotto posted an article about raised beds for veg growing - that's not exactly what you want for most perennials. What you intend growing does affect what growing medium you use, but assuming you are growing things like lavender, salvias, verbena and ornamental grasses, your mix doesn't have to be overly rich.
Personally I'd go for 20cm depth of sharp sand, overlaid with a 35cm depth 2:1 mix of topsoil and compost.
"What is hateful to you, do not do to your neighbour".
The angular nature of the beds suggests that a measure of formal planting would be appropriate. For height I would go for standard bay or olive trees (standard = clear stem with a blob of green on top), or maybe a standard holly tree. Underneath these I would plant erigeron karvinskianus Profusion, a daisy like flower that will bloom all summer.
I would have banks of tall, statuesque alliums such as Purple Sensation but their strap like leaves look very tatty and need a plant to cover them. Geranium Rozanne is a good choice. Another bulb (allium is a bulb) that I would consider is camassia leichtnii, a blue flower just a little like an elegant gladiolus. These like moist conditions so put them in your shadiest spot. If you wanted to grow agapanthus - and I would - I would ensure its growing space has a mix of 50% regular soil and 50% sharp sand or grit. Agapanthus Midnight Blue is a favourite of mine. Bearded irises would look splendid in raised beds and there is a myriad of colours so you’d have no difficulty finding one to work with your colour scheme. Irises and agapanthus are happy in baking sun.
In a formal planting scheme lots of just a few plants will look much more effective than a dolly mixture of plants in haphazard ones and twos.
PS In your third photo the two blue flowers on long stalks in the foreground are agapanthus. On the bed by the fence the purple plant is one of the many salvias. They would be another great choice. In the middle photo behind the white roses (?) I see lavender, very probably the popular Hidcote variety.
Just to add the following graphic (for green roofs but applicable here) - it's generally not recommended to much deeper than 350mm with your topsoil. Where it refers to 'mineral sub-substrate' it means something like subsoil, sand, or gravel, which contains no organic material. So you were on the right lines with your 150mm of gravel. I think sharp sand is slightly better though, as it has some water retention capacity, while still being free draining.
"What is hateful to you, do not do to your neighbour".
I’d suggest going for deeper topsoil in any area where you will plant shrubs, as the width is not great and most shrubs would need a planter considerably larger to cope for more than two or three seasons. Topsoil is the best medium to use for the shrubs in particular, and also for most perennials and grasses—you’ll only want compost for tender bedding plants really.
In terms of what to plant, grasses that move with the breeze but don’t collapse will be good, like Stipa tenuissima, Panicum virgatum or Pennisetum ‘Hameln’.
For the corner standards or trimmed shrubs, box is still worth a try. Ilex crenata won’t like the periods of drought in this kind of situation, but there are nice little Ligustrum standards you can get and a tough shrub for clipping would be Lonicera nitida.
For perennials and infill, you probably are best off aiming for things that don’t have too much of an ‘off’ season, since they are raised up and will be on show all the time. Hardy geraniums include some brilliant long-flowering forms in blues and pinks; check out The Plantsman’s Preference or Dorset Perennials website for good examples. Helianthemum, Cistus and Sage varieties that aren’t too tall will all be good. Crocosmia will add good sharp foliage and warm colours, and Sedums/Hylotelephium are also great. Bracketing plants like Erigeron glaucus, Bergenias and Ceratostigma plumbaginoides will also serve your needs well.
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Bit concerned that your bed is very narrow so you might want to ask about what to plant before you get soil.
If you can only grow smaller planrs they need less soil depth so you could vary the depth, maybe deeper in the corners for a shrub or small tree.
Maybe with some buxus balls.
I can see grasses in there, festuca is a lovely evergreen one but quite short.
There is some agapanthus, which doesnt mind being restricted.
At the back in your last photo the mauve plant might be a salvia.
With the white flowers no idea what they are but maybe ground cover roses, there seems to be lavender in there too.
Personally I'd go for 20cm depth of sharp sand, overlaid with a 35cm depth 2:1 mix of topsoil and compost.
https://www.gardenersworld.com/plants/alternatives-to-box/
The angular nature of the beds suggests that a measure of formal planting would be appropriate. For height I would go for standard bay or olive trees (standard = clear stem with a blob of green on top), or maybe a standard holly tree. Underneath these I would plant erigeron karvinskianus Profusion, a daisy like flower that will bloom all summer.
I would have banks of tall, statuesque alliums such as Purple Sensation but their strap like leaves look very tatty and need a plant to cover them. Geranium Rozanne is a good choice.
Another bulb (allium is a bulb) that I would consider is camassia leichtnii, a blue flower just a little like an elegant gladiolus. These like moist conditions so put them in your shadiest spot. If you wanted to grow agapanthus - and I would - I would ensure its growing space has a mix of 50% regular soil and 50% sharp sand or grit. Agapanthus Midnight Blue is a favourite of mine. Bearded irises would look splendid in raised beds and there is a myriad of colours so you’d have no difficulty finding one to work with your colour scheme. Irises and agapanthus are happy in baking sun.
In a formal planting scheme lots of just a few plants will look much more effective than a dolly mixture of plants in haphazard ones and twos.
PS In your third photo the two blue flowers on long stalks in the foreground are agapanthus. On the bed by the fence the purple plant is one of the many salvias. They would be another great choice. In the middle photo behind the white roses (?) I see lavender, very probably the popular Hidcote variety.
https://www.classicroses.co.uk/roses/ebb-tide-bush-rose.html
For the corner standards or trimmed shrubs, box is still worth a try. Ilex crenata won’t like the periods of drought in this kind of situation, but there are nice little Ligustrum standards you can get and a tough shrub for clipping would be Lonicera nitida.