May and June is the snow melting season, rivers are high, the ground is boggy. Therefore, you shouldn't start before July unless you plan to do road walking.
Best time for hiking in Scandinavia is August til beginning of September - which of course would not work for a thru-hike. But I wouldn't recommend a thru-hike anyway on the E1 in Scandinavia, you should better make it within 2 or 3 summers.
It is possible, but it's no fun... Sven Schoppe (thelongwalk.de) started in May and had to use the roads for the first weeks and later still had problems with boggy ground and snow storms. Daniel Alvarez (predictablylost.com) started in June and he really made it but it didn't look like much fun in his videos. Last year, there were two German bloggers who tried it (separately) but they also started too early and had to skip the Northern part and got back on the track further south resp. cancel the tour.
So my advice: unless you're not a very experienced hiker, don't start before July. You might make it from Nordkapp til Grövelsjön from start of July until mid-September. For the rest of the Scandinavian section, you should wait for the following year and start again mid-June in Grövelsjön.