I've flown Qatar Qsuite & their standard business class and BA's Club World.
Check in:
You get dedicated check ins for flying business class with both airlines and fast track security. Neither is superior to the other.
Lounge:
The BA lounge at T5 is good. Excellent selection of hot buffet food, a plethora of alcohol and plenty of seating. I love the room with the TV and chairs... feels sort of hidden and because it's dark and you can't really walk through there to get somewhere else in the lounge, I felt like it was easier to nod off! (I haven't used the business class lounge in a long time so unsure of whether the chairs still exist). Depending on which day you fly and what time - the showers can be a nightmare to book. You often have to wait 30 minutes before one is available so be sure to book one as soon as you enter the lounge if you want to use it. You also get access to the Arrivals Lounge if your flight lands before 11am. Lovely way to freshen up as they have showers in there and you have the opportunity to get a hot breakfast if you've missed breakfast and are hungry - this is all included in the price of your ticket.
The Qatar lounge in comparison is amazing. The ambiance is phenomenal! You can get a deli style bar (however they serve you) or dine at a table - your choice. The lounge is located right next to the gate so you don't have to venture far. However Qatar have recently changed their lowest business class fare which alas no longer includes access to the lounge so bear that in mind when choosing. You could always pay extra for a 3rd party lounge.
The plane: BA use an A380 on the LHR-SIN route which is the largest passenger aircraft. Qatar use a mixture of 777, A380s and A350s from London to Doha and A350s to Singapore - which are all twin engine planes. If you're a nervous flyer - I'd recommend choosing flights with the A380 as turbulence won't be felt as bad (the Bay of Bengal is notorious for turbulence 
The seat: Qsuite is by far the best business class I've flown on. It is excellent in every way. You can your own private "suite" and direct aisle access at every seat. You get The White Company toiletries, PJs (you get to keep them) and bedding (which forced me to buy new linen from them when I got home
. I don't like the way your feet are restricted in the footwell as I am a side sleeper and found it annoying to sleep while my partner sleeps on his back and had no problems. My partner and I slept in the middle seats which is the best for couples as we turned our seats into a bed! The crew put on our mattress topper and made it up while we went to the loo to change into our PJs. I would definitely recommend these seats as you're flying as a couple.
BA on the other hand, you do not have aisle access at every seat and the seat is very dated. You have to climb over someone depending on where you sit. What I love about the seat is though, it is very comfy. I'd say it's better to sleep on than Qsuite and you aren't restricted in which way to sleep. You don't get PJs in business however you do get a mattress topper which you have to put on yourself which isn't really a problem. The White Company provides the bedding as well.
Food and entertainment:
Qatar: the food is excellent. You get a four course meal. The crew go seat-to-seat taking orders while boarding once you're settled into seat and have a look at the menu. You also get to dine on demand which is very rare for business and I loved it. The food varies on the route but what I had was excellent.
BA: BA does not offer dine on demand but similar, you get a four course meal which is excellent. You get given a menu before take off and the crew take your order while doing the trolley service.
Crew: both are friendly.
I know this is quite long but I miss flying and I feel like this was a good way to bring back memories! BA are currently refitted all of their planes with their new business class so my input might be out of date by the time you fly with them (if you choose to do so). Bear in mind - you have to make a layover in Doha in the middle of the night which can be annoying if you want to sleep straight through the flight to Singapore.
I would definitely try to fly business, it's a long flight and it's so worth it. I no longer want to fly economy ever again so I find ways to avoid it
give it a try! Singapore is a great destination. I would choose Qatar overall.
TLDR; both are good however I'd rather fly Qatar despite the middle-of-the-night layover. It's an excellent airport to transit in and you get lounge access there to shower.