I'm in a broadly similar position to the OP, looking into the industry from the outside.
Firstly, make sure you can get a Class 1 medical, you won't be able to fly commercially without one.
As has been said, apply for any fully sponsored schemes which come up. An important part of these is that the airline has skin in the game, so to speak. It's not in their interest to terminate the programme mid-way through due to an industry downturn or similar. If it's a self-funded "tagged" programme, the airline doesn't have the same investment in making sure you have a job at the end.
As for the self-funded full-time/part-time debate, how easily can you return to your current career after a break? If you can pick up contractual ad-hoc work, full-time training might be worth a bet, however if it's tough to get back in then you might be better going part-time. Commercial aviation is very cyclical when it comes to employment and it can change very quickly. The job market may be strong now but will it be in two years' time? No-one really knows. I'm a bit risk-averse and tend to plan around the worst outcome and work from there but that's just me.
You haven't said how much flying you've done so far, it may be worth taking up gliding or getting a PPL to see if you enjoy it. The only thing to watch out for is that TUI won't take you on their sponsored scheme if you have over 100 hours.