Should I switch to a smaller chain ring? I am back to cycling after 7 years absence due to injury and time commitments. When I started 3 months ago I weighed 20kg more (105 kg in total) than when I stopped. My target is to build fitness (100+ miles per week) and lose the weight gradually (1kg per month) both of which I currently reach. 

My bike has a standard Shimano 105/5800 groupset with 53/39 chain rings and 11-32 cassette. 

In the flats with a little wind from the back I tend to be in the largest gears whilst running at a decent cadence, but as soon as I start climbing more than 3% (and I live in a hilly area) I tend to be on the smallest of gears. 

So my real question. Would it make sense for me to switch to the smaller 50/34 chain rings? Or is it more sensible to tough it out for now and expect that both the weight reduction and fitness improvement will get me into a more comfortable position relatively quickly.
If I were in your situation, I would definitely get a compact (50/34) crank.  Gaining lower gears for climbing makes some uphills possible and other uphills easier. All you have to give up is maybe some top speed on downhills. You're still able to ride those downhills, only you'll be going 55kph instead of 60kph.  Or increase your cadence.  In general I come across a lot of people who think they need huge gears because they somehow want to have a really low cadence.

Since you said you're riding for fitness and weight loss, I don't think that top speed is a priority for you -- you're not competing against others in a race.  It's much more important that you can handle hills and don't hurt your knees.

My top gear is 46/11 and I never find that to be a limitation.  A cadence of 120rpm is quite reasonable for a few minutes (few descents are really long) and 120rpm with a gear of 46/11 is approximately 65kph.  With a top gear of 50/11, it's 71kph.