That's curious, because it always used to be the case that new entrants with GTR (inc. Thameslink, Southern etc.) received free travel on Go-Ahead services from the first week of joining. When you say "midlands", you are thinking of London Midland, which is a Go-Ahead rail company, and as such covered by this. GTR in itself is a Go-Ahead company, too. The general rule of thumb with GB train operating companies (TOCs) is that you receive free travel on the services run not just by your company, but also on franchised rail services provided by the owning group as a whole.
I suspect the documentation which mentions these limitations of free travel is incorrect, and you will find out more accurate information when you've joined. In any case, GTR staff are usually issued with a staff "key" card as ID and for travel, which has printed text on the back which explicitly states that it entitles the holder to free travel on Go-Ahead rail services, not just GTR itself, which is only one of the providers of such services.
At certain times you may find GTR's Southern-branded services operate through Hemel Hempstead, in which case you would be able to use those without quibble anyway, as it's well within the same company. As for LM, it's unlikely you would have any trouble when presenting a GTR / Thameslink ID, but bear in mind all their services have conductors, so you could try asking before boarding on the first few occasions.
What may well be limited during your training is the ability to get and use a Priv card (ATOC Privilege Rate), which gives you a 75% discount on most of the TOCs you'd need to buy a ticket to use, as well as a few other benefits, such as free use of Croydon Tramlink. Unless you're heading much further afield, it's unlikely this would cause great issues for your commuting. The same could apply for the PTAC card (for 75% off Tube / DLR on Oyster) and there would definitely be limitations for FIP (European travel benefits).
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