We moved back to the UK at the beginning of June 2017. I am now trying to renew my British license, that has expired, online and it is saying that I need to do the paper version of the application. Should I renew my UK one or exchange my Canadian license?

Have you contacted the DVLA?

Some of our members had lost their UK licence...but the DVLA had been able to trace it and issue a new one.

I haven't lost it, but I don't know if I should be exchanging Canadian for GB it or renewing my old GB one that I but is expired.

FWIW I would renew...but that's only my opinion.

The is no British license, it is a licence. :-)
licence - noun, license - verb.

British driving licence does not expire until 70 years of age. What does expire is the photocard. Renew it at one of the main Post Offices that takes identification photos.
Branch Finder | Post Office
Enter your postcode, tick the box "DVLA photocard renewal" and click on search.

A couple of things to unpack here. Since you still have your GB licence did you pass your Canadian driving test separately? Is it a UK photocard licence that you hold (and has expired?)

You You are required to exchange your DL for a Canadian DL. In which case there is no test.

Except the OP appears to have both hence my question.

I believe you are legally required to hand over all previous DL's...in exchange for a Canadian DL. Of course you could hand over the pink photo card (which many think is a DL)...and keep the actual paper DL.

Depends when the OP moved to Canada, & possibly which province they moved to.... when I moved in 2001 (to Ontario), UK licences were not accepted for "exchange", & I had to sit both a written & practical test. The rules have since changed. Guess different provinces may have their own rules...

We are in the process of planning a move back to the UK, & I recently called DVLA with this question about my expired UK licence. I gave the person who answered my driving licence#, & she said that I should just apply to renew it, & also pass a medical for my LGV entitlement, & it'd be good to go. While waiting for the new licence to arrive I'd be good to drive cars only. I was given form #s to use.... not sure of an online service, that wasn't presented as an option for me.

4 years ago we exchanged our New Jersey DL for an Ontario DL.

Right. Seem to remember when I was getting my ON licence, there was a short list of countries that were accepted for exchange, IIRC USA, Germany, Switzerland, S Korea & Japan? maybe more. US licences were fully transferable with all classes of vehicles, but the other countries could only exchange car entitlement. UK was added to the list shortly afterwards.
They did want to see my UK licence, as previous driving experience meant that I only had to sit the final test, instead of starting as a beginner & going through the whole multi-year graduated process.
Similarly, when my wife & I were in the UK she was able to exchange her ON licence for a UK one, including for vehicles with manual shifting. This has now changed, & it is only possible to exchange a Canadian licence for UK licence for automatic transmission vehicles, unless evidence can be shown that the test was passed in an automatic.

Also similar here. Back in 2006 we exchanged our Illinois licenses for Ontario. However as we found out, it's not seamless for every state. Illinois is one of the few states that does not have an electronic agreement with Ontario for verifying motor vehicle violations. That meant we had to print the official Illinois violation report for each of us and submit that in person at the Ontario service centre office.

Otherwise no exchange possible. So something to check before going in.

That reminds me...around 17 years ago our daughter applied for an Ontario DL. She had to obtain a driving transcript from New Jersey...they were one of the few US states who didn't have an agreement with ON to do it electronically. We spent ages running around to various DMV offices trying to get hold of it. My husband tells me we also needed the transcripts...but they were much easier to obtain 13 years later.

Apologies to the OP...I seem to have taken the thread off topic.