If you’re told that your software team needs to increase delivery speed but you can’t spend budget on new tooling or platforms, where would you make improvements?

One avenue that could help is to agree on an initial scope and prohibit and change request unless approved by a design authority kind of council.

For me, one of the way to increase delivery speed is keeping a fixed scope.  We've experienced too many times projects where numerous change request just pushed back the delivery date.   

And most of the time, when it comes from Line of Business stakeholders, it appears to be so legitimate that few will question it.

I would take three immediate actions:
1) Clarify the "need to increase delivery speed", what are you being told, and is it clear enough?  "Faster" is generally a poor mandate, and you'll need clarity of expectations to form a plan, vision and strategy for your team.  

2) Establish measurements (comprehensive).  No standard metrics exist but that does not mean you can't measure anything.  Look at the DORA key metrics, examine your end-to-end SDLC, and see if you can determine how long a typical feature takes from idea to delivery, where it stalls out (where are the bottlenecks).  

You will NOT be able to move faster unless you know how fast you're moving, where time and effort is being spent and why.

3) Develop a vision and strategy to address what you think are the most significant bottlenecks.  Delivery your vision and strategy, with timelines upwards.  Explain the metrics you've found in #2, and how your changes will adjust them over time towards new levels.

Once your vision and strategy are supported and aligned with what is being asked of you, take it to the team.  Explain the need to change, the vision (what is not working, what we will change) and the strategy (here's how we will get the changes done).

.....

Skipping #1 or #2 will only cause yourself trouble and you'll only end up frustrating the team because if you lack clarity on what "faster" means, and how to measure it, they'll just be more in the dark.