1. Do bigger European airlines still consider pilots who went the modular route?
Definitely. Ryanair, Jet2 and Flybe to name a few.
2. Will airlines still consider me, even if I did 'fail' half of my ATPLs at first attempt all those years ago?
Good question, but I thought ATPL results 'expire' after a few years if you haven't finished the course? But to answer the question, it depends on the airline as some may be picky. I believe Ryanair allow 2 fails, but I am basing that on something I heard a few months ago. Note that ATPL fails are merely a form of reducing the CV pile size. A bad set of ATPL results does not mean that the candidate is a bad pilot as I've heard there's a lot of crap needed to be learnt as well as ambiguous exam questions.
I would never let that be a deciding factor as to whether or not to start training. If you are determined and have the passion then worry not, you will eventually be flying for a living. It may not be within a week, month or year after graduating but if you stay within aviation and don't give up then you will eventually find yourself in the RHS.
3. Are the theoretical elements possible through distance learning (e.g. ground school)?
Yes, e.g. BGS. Search around PPRune to find lots of people having done their ATPL while married with kids and working a full time job. A lot of them are also now flying.
4. Is modular the way to go in my situation? Any other route that can be done ‘part-time'?
Pretty sure most integrated schools are full time. So if you want to do it all part-time then modular would definitely be the way. With the current pilot forecast for the next 2 decades it seems like pilots will be in much greater demand. Some people have mentioned that there has been a decrease in demand over the past 12 months but it could be explained by the sudden supply of pilots from Monarch/Air Berlin.
5. What schools are recommended within easy reach of London (especially for the PPL)? I’m willing to travel up to 2 hours by train - no problem - so long as the school is reputable and my qualifications respected as I continue doing the licences/courses.
6. Any other input is much appreciated 
I got my PPL a few years ago and starting my ATPL in summer after graduating from University. Would be happy to keep in touch with you or anyone else who reads this so feel free to shoot me a PM. We all have the same goal here.