Hi!
All 1.*.* versions of H2 are entirely outdated and aren't supported in any way.
1. H2 1.4.* has two different storage engines.
2. H2 1.4.199 and older versions have different synchronization models for every engine (1.4.200 uses multi-threaded mode for MVStore and single-threaded mode for PageStore unconditionally).
3. H2 1.4.197 and older versions can use either table-level locks or row-levels locks when legacy PageStore engine is used (since 1.4.198 PageStore uses table-level locks unconditionally).
So possibility of concurrent execution depends on your setup. Most likely you use a legacy PageStore engine and therefore database works in single-threaded mode with table-level locks.