yes, he is probably correct that a new wheel is in order. if you really like the bike and are planning to keep it for yourself as a regular rider you can cut out that hub and get a new wheel built on it. Or you can search around and hope to find another vintage 3 speed for cheep that has a wheel intact. the nice thing about building a new wheel is you can get aluminum which will be much lighter and stop much better in wet conditions.
You can always replace only the broken spokes, it's not like it's impossible in any way. However, spoke breakage is often due to spokes being fatigued due to being poorly tensioned originally. This means that the remaining life in the other spokes is questionable at best. Still, if you ca deal with spoke keeping popping every now and then it's certainly doable.
The sensible advise for a reliable ride OTOH is to replace them all. If the rim is worn, or if you want another one, this is a good opportunity. As bigdaddy suggests, alloy rims offer better braking if you're using rim brakes. Still, braking would improve more if you replaced the front rim.
Assuming you can keep the rim, a relace(spokes + labor) shouldn't be 60£, not even in London.
THen again, if the rim is OK it can be relaced with new spokes. Plenty of posters here who've tackled that by themselves. It is possible to replace them one by one, although it is tedious and more time consuming that doing it all in one go. For many rookie mechanics that approach feels less daunting. Being an IGH hub you either already have clearance for spoke replacement, or the sprocket will be very easy to remove.
the LBS guy is correct. 3 broken spokes makes it likely that many or most of the spokes are fatigued and on their way to breakage (whereas with 1 broken spoke there is more chance it's just a fluke).
However, if you can replace the three spokes, and then properly true and tension the wheel, you might be able to get away without more breakage, for a long time. (bringing the spokes up to proper tension makes them less likely to break, even if they are already on the road to fatigue breakage).
You could even replace the spokes yourself, and work on bringing the wheel up to proper true/tension. Sheldon Brown's instructions may be helpful to you here: http://sheldonbrown.com/wheelbuild.html
On the other hand, if the bike is in good condition otherwise, paying for a wheelbuild (and getting an inexpensive aluminum rim to replace the steel) isn't out of the question.
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+1, replace all the spokes.
Now would be a good time to consider replacing the rim as well. Replacing the rim is probably not necessary, but would have various advantages already mentioned (better braking; lighter; and the new rim will be round, whereas your old one may have issues).