There are also red and white currants, gooseberries, strawberries, tayberries and of course blueberries, the original woodland berry. Alpine strawberries would be good, too but love a bit more sun.
Currants/rhubarb/strawberries love sunshine, raspberries do well in partial shade and blueberries can cope with partial shade to shade just fine.
If birds are a problem you are better off netting them as soon as the fruits form. Squirrels might be a problem, too. You can buy fancy fruit cages and or just make your own using some battens and netting. Ground pegs to hold the netting down.
Most will crop the year after planting but will get better after a year or so. There are summer and autumn fruiting raspberries and they are pruned at different times, so make a note of which type you plant where.
Personally, I don't think there is too much work involved in any of them. Rhubarb is no work at all, blueberries are small and are being picked individually, so are hard work at the picking stage. Could be outweighed by the fact that they are £££ to buy. My priority would be to grow what I like, so if you don't like rhubarb there's little point in growing it even if it is zero effort.
Currants and gooseberries need to be pruned correctly but it is not difficult.
Raspberries will need support (post and wire arrangement plus netting) but once in place it will be good for a few years. If you use hardwood and nylon wire/gripples it will last the lifetime of the raspberry canes (ca 10 years).
Strawberries need replacing every three years or so but you can save runners. You also need to give them a new bed when replanting to avoid disease/virus.
Also recommend you test the soil to establish pH levels which may have a bearing on which soft fruits will do better in your location.
HTH to get you planning.