Hi Muppet
The nature of the interactions with your family which you've posed here appear to be overwhelmingly positive, so let us stay with this as the overriding thought.
I can fully understand your mother's concerns. Gay is easy these days, but being trans places a much bigger burden on families in terms of adaptation and acceptance. Both of these things take time. I suspect that your family may be more intellectually accepting of you as a transgender person than they are emotionally accepting, and again, this is a process. The thing is, you've lived this process, but they've just recently been thrust into it, with no point of reference or real suspicion. It requires some degree of patience and empathy from you, and from them.
I can also understand why your family is not offering to help pay for your transition costs. You appear to have been fairly well-adjusted, and not in any obvious and constant distress as a person, so, again, from your mother's perspective, it's not as if fixing your gender will cure you of a life of emotional torture. And taking this into account, it is probably very hard for a parent under such circumstances to want to play an active part in changing you into someone who they think will be a totally different person, and not necessarily a happier or better adjusted one.
I am giving you this viewpoint because I think we transgender people frequently expect our families to just roll over and accept what we are saying. A bit of skepticism is probably healthy, as it may make you focus on what you really want from your life and whether now is the time to try achieve it. For example, I deferred the decision for 20 years, and made sure that those intervening years were well spent.
My key messages to you are to keep patient and remain consistent. Now that you've dropped your bombshell, make a complete plan for what you want to do next. Analyse the costs, time, risk etc. Do this for your whole life, not just for the transition part, and produce a plan that identifies the risks and proposes ways to mitigate them.
You're doing the right things so far, in my opinion.
Regards
Julia