@L1234
There used to be (and probably still are) some duties that are 'pass' to Blackfriars, St Albans, or Luton at the start - you may well find you'll be able to travel straight there and sign on there. Each of those locations should have a signing on sheet and a late notice case. Certainly St Albans and Luton as they are traincrew depots, but I believe the facility is there at Blackfriars as well.
You just need to agree with the Duty Resource Manager in advance and phone up at your booking on time to say all is going to plan.
Also: At Thameslink you get paid from the platform to walk back to the signing on point at the traincrew depot at the end of your shift. At Cricklewood this is a long walk away and your booking off time is about half an hour after you get relieved in the platform. Particularly if you were driving a southbound train, you'd stay on it and be a good way nearly home by the time of your booking off time!
On an early afternoon shift this would work well and you could get the train in and back home again.
I don't know where Orpington and Gillingham are in relation to you, but putting in for a transfer might be something to think about if they are nearer. (I think it's two years productive driving, or three years since start date, until you can go on the transfer list.)